diff --git a/src/topologies/halfedge.jl b/src/topologies/halfedge.jl
index ade856472..b0cd7b220 100644
--- a/src/topologies/halfedge.jl
+++ b/src/topologies/halfedge.jl
@@ -144,126 +144,119 @@ function HalfEdgeTopology(elems::AbstractVector{<:Connectivity}; sort=true)
 
   # sort elements to make sure that they
   # are traversed in adjacent-first order
-  adjelems = sort ? adjsort(elems) : elems
-  eleminds = sort ? indexin(adjelems, elems) : 1:length(elems)
+  elemsort = sort ? adjsort(elems) : elems
+  eleminds = sort ? indexin(elemsort, elems) : 1:length(elems)
+  adjelems::Vector{Vector{Int}} = map(collect ∘ indices, elemsort)
 
   # start assuming that all elements are
-  # oriented consistently as CCW
-  CCW = trues(length(adjelems))
+  # oriented consistently (e.g. CCW)
+  isreversed = falses(length(adjelems))
 
-  # initialize with first element
+  # initialize half-edges using first element
   half4pair = Dict{Tuple{Int,Int},HalfEdge}()
-  elem = first(adjelems)
-  inds = collect(indices(elem))
-  v = CircularVector(inds)
-  n = length(v)
-  for i in 1:n
-    half4pair[(v[i], v[i + 1])] = HalfEdge(v[i], eleminds[1])
+  elem = first(eleminds)
+  inds = first(adjelems)
+  n = length(inds)
+  for i in eachindex(inds)
+    u = inds[i]
+    v = inds[mod1(i + 1, n)]
+    # insert halfedges u -> v and v -> u
+    he = get!(() -> HalfEdge(u, elem), half4pair, (u, v))
+    half = get!(() -> HalfEdge(v, nothing), half4pair, (v, u))
+    he.half = half
+    half.half = he
   end
 
-  # insert all other elements
-  for e in 2:length(adjelems)
-    elem = adjelems[e]
-    inds = collect(indices(elem))
-    v = CircularVector(inds)
-    n = length(v)
-    for i in 1:n
-      # if pair of vertices is already in the
-      # dictionary this means that the current
-      # polygon has inconsistent orientation
-      if haskey(half4pair, (v[i], v[i + 1]))
-        # delete inserted pairs so far
-        CCW[e] = false
-        for j in 1:(i - 1)
-          delete!(half4pair, (v[j], v[j + 1]))
-        end
-        break
+  # update half-edges using all other elements
+  added = false
+  disconnected = false
+  remaining = collect(2:length(adjelems))
+  while !isempty(remaining)
+    iter = 1
+    while iter ≤ length(remaining)
+      other = remaining[iter]
+      elem = eleminds[other]
+      inds = adjelems[other]
+      n = length(inds)
+      if anyhalf(inds, half4pair) || disconnected
+        # at least one half-edge has been found, so we can assess
+        # the orientation w.r.t. previously added elements/edges
+        added = true
+        disconnected = false
+        deleteat!(remaining, iter)
       else
-        # insert pair in consistent orientation
-        half4pair[(v[i], v[i + 1])] = HalfEdge(v[i], eleminds[e])
+        iter += 1
+        continue
       end
-    end
 
-    if !CCW[e]
-      # reinsert pairs in CCW orientation
-      for i in 1:n
-        half4pair[(v[i + 1], v[i])] = HalfEdge(v[i + 1], eleminds[e])
+      # if the other element is already claimed by any half-edge
+      # then the element must be reversed before further updates
+      isreversed[other] = anyhalfclaimed(inds, half4pair)
+
+      # insert half-edges in consistent orientation
+      if isreversed[other]
+        for i in eachindex(inds)
+          u = inds[i]
+          v = inds[mod1(i + 1, n)]
+          he = get!(() -> HalfEdge(v, elem), half4pair, (v, u))
+          if isnothing(he.elem)
+            he.elem = elem
+          else
+            assertion(
+              he.elem === elem,
+              lazy"duplicate edge $((v, u)) for element $(elem) is inconsistent with previous edge $he"
+            )
+          end
+          half = get!(() -> HalfEdge(u, nothing), half4pair, (u, v))
+          he.half = half
+          half.half = he
+        end
+      else
+        for i in eachindex(inds)
+          u = inds[i]
+          v = inds[mod1(i + 1, n)]
+          he = get!(() -> HalfEdge(u, elem), half4pair, (u, v))
+          he.elem = elem # this may be a pre-allocated half-edge with a nothing `elem`
+          half = get!(() -> HalfEdge(v, nothing), half4pair, (v, u))
+          he.half = half
+          half.half = he
+        end
       end
     end
-  end
 
-  # add missing pointers
-  for (e, elem) in Iterators.enumerate(adjelems)
-    inds = CCW[e] ? indices(elem) : reverse(indices(elem))
-    v = CircularVector(collect(inds))
-    n = length(v)
-    for i in 1:n
-      # update pointers prev and next
-      he = half4pair[(v[i], v[i + 1])]
-      he.prev = half4pair[(v[i - 1], v[i])]
-      he.next = half4pair[(v[i + 1], v[i + 2])]
-
-      # if not a border element, update half
-      if haskey(half4pair, (v[i + 1], v[i]))
-        he.half = half4pair[(v[i + 1], v[i])]
-      else # create half-edge for border
-        be = HalfEdge(v[i + 1], nothing)
-        be.half = he
-        he.half = be
-      end
+    if added
+      added = false
+    elseif !isempty(remaining)
+      disconnected = true
+      added = false
     end
   end
 
-  # save halfedges in a vector of pairs
+  # add missing pointers and save halfedges in a vector of pairs
   halves = Vector{Tuple{HalfEdge,HalfEdge}}()
   visited = Set{Tuple{Int,Int}}()
-  for ((u, v), he) in half4pair
-    uv = minmax(u, v)
-    if uv ∉ visited
-      push!(halves, (he, he.half))
-      push!(visited, uv)
-    end
-  end
-
-  HalfEdgeTopology(halves; nelems=length(elems))
-end
-
-function adjsort(elems::AbstractVector{<:Connectivity})
-  # initialize list of adjacent elements
-  # with first element from original list
-  list = indices.(elems)
-  adjs = Tuple[popfirst!(list)]
+  for (e, inds) in enumerate(adjelems)
+    inds = isreversed[e] ? circshift!(reverse!(inds), 1) : inds
+    n = length(inds)
+    for i in eachindex(inds)
+      u = inds[i]
+      v = inds[mod1(i + 1, n)]
+      w = inds[mod1(i + 2, n)]
 
-  # the loop will terminate if the mesh
-  # is manifold, and that is always true
-  # with half-edge topology
-  while !isempty(list)
-    # lookup all elements that share at least
-    # one vertex with the last adjacent element
-    found = false
-    vinds = last(adjs)
-    for i in vinds
-      einds = findall(e -> i ∈ e, list)
-      if !isempty(einds)
-        # lookup all elements that share at
-        # least two vertices (i.e. edge)
-        for j in sort(einds, rev=true)
-          if length(vinds ∩ list[j]) > 1
-            found = true
-            push!(adjs, popat!(list, j))
-          end
-        end
+      # update pointers prev and next
+      he = half4pair[(u, v)]
+      he.next = half4pair[(v, w)]
+      he.next.prev = he
+
+      uv = minmax(u, v)
+      if uv ∉ visited
+        push!(halves, (he, he.half))
+        push!(visited, uv)
       end
     end
-
-    if !found && !isempty(list)
-      # we are done with this connected component
-      # pop a new element from the original list
-      push!(adjs, popfirst!(list))
-    end
   end
 
-  connect.(adjs)
+  HalfEdgeTopology(halves; nelems=length(elems))
 end
 
 paramdim(::HalfEdgeTopology) = 2
@@ -338,3 +331,67 @@ nfacets(t::HalfEdgeTopology) = length(t.halfedges) ÷ 2
 # ------------
 
 Base.convert(::Type{HalfEdgeTopology}, t::Topology) = HalfEdgeTopology(collect(elements(t)))
+
+# -----------------
+# HELPER FUNCTIONS
+# -----------------
+
+function adjsort(elems::AbstractVector{<:Connectivity})
+  # initialize list of adjacent elements
+  # with first element from original list
+  list = indices.(elems)
+  adjs = Tuple[popfirst!(list)]
+
+  # the loop will terminate if the mesh
+  # is manifold, and that is always true
+  # with half-edge topology
+  while !isempty(list)
+    # lookup all elements that share at least
+    # one vertex with the last adjacent element
+    found = false
+    vinds = last(adjs)
+    for i in vinds
+      einds = findall(e -> i ∈ e, list)
+      if !isempty(einds)
+        # lookup all elements that share at
+        # least two vertices (i.e. edge)
+        for j in sort(einds, rev=true)
+          if length(vinds ∩ list[j]) > 1
+            found = true
+            push!(adjs, popat!(list, j))
+          end
+        end
+      end
+    end
+
+    if !found && !isempty(list)
+      # we are done with this connected component
+      # pop a new element from the original list
+      push!(adjs, popfirst!(list))
+    end
+  end
+
+  connect.(adjs)
+end
+
+function anyhalf(inds, half4pair)
+  n = length(inds)
+  for i in eachindex(inds)
+    uv = (inds[i], inds[mod1(i + 1, n)])
+    if haskey(half4pair, uv)
+      return true
+    end
+  end
+  return false
+end
+
+function anyhalfclaimed(inds, half4pair)
+  n = length(inds)
+  for i in eachindex(inds)
+    uv = (inds[i], inds[mod1(i + 1, n)])
+    if !isnothing(get(() -> HalfEdge(0, nothing), half4pair, uv).elem)
+      return true
+    end
+  end
+  return false
+end
diff --git a/test/topologies.jl b/test/topologies.jl
index 3e51a0b7e..7158330a7 100644
--- a/test/topologies.jl
+++ b/test/topologies.jl
@@ -372,8 +372,15 @@ end
     for e in 1:nelements(topology)
       he = half4elem(topology, e)
       inds = indices(elems[e])
-      @test he.elem == e
-      @test he.head ∈ inds
+      for _ in inds
+        @test he.elem == e
+        @test he.head ∈ inds
+        @test he.next.elem == e
+        @test he.prev.elem == e
+        @test he.next.prev == he
+        @test he.prev.next == he
+        he = he.next
+      end
     end
   end
 
@@ -483,6 +490,7 @@ end
   # correct construction from inconsistent orientation
   e = connect.([(1, 2, 3), (3, 4, 2), (4, 3, 5), (6, 3, 1)])
   t = HalfEdgeTopology(e)
+  test_halfedge(e, t)
   n = collect(elements(t))
   @test n[1] == e[1]
   @test n[2] != e[2]
@@ -492,14 +500,19 @@ end
   # more challenging case with inconsistent orientation
   e = connect.([(4, 1, 5), (2, 6, 4), (3, 5, 6), (4, 5, 6)])
   t = HalfEdgeTopology(e)
+  test_halfedge(e, t)
   n = collect(elements(t))
-  @test n == connect.([(5, 4, 1), (6, 2, 4), (6, 5, 3), (4, 5, 6)])
+  @test n == connect.([(4, 1, 5), (4, 6, 2), (6, 5, 3), (4, 5, 6)])
+
+  e = connect.([(1, 2, 3), (1, 3, 4), (2, 5, 3), (5, 4, 6), (3, 5, 4)])
+  t = HalfEdgeTopology(e)
+  test_halfedge(e, t)
 
   # indexable api
   g = GridTopology(10, 10)
   t = convert(HalfEdgeTopology, g)
-  @test t[begin] == connect((13, 12, 1, 2), Quadrangle)
-  @test t[end] == connect((110, 121, 120, 109), Quadrangle)
+  @test t[begin] == g[begin]
+  @test t[end] == connect((120, 109, 110, 121), Quadrangle)
   @test t[10] == connect((22, 21, 10, 11), Quadrangle)
   @test length(t) == 100
   @test eltype(t) == Connectivity{Quadrangle,4}
diff --git a/test/toporelations.jl b/test/toporelations.jl
index 4b158ade1..1afbe48e9 100644
--- a/test/toporelations.jl
+++ b/test/toporelations.jl
@@ -445,30 +445,30 @@ end
   elems = connect.([(1, 2, 3), (4, 3, 2)])
   t = HalfEdgeTopology(elems)
   ∂ = Boundary{2,0}(t)
-  @test ∂(1) == (2, 3, 1)
+  @test ∂(1) == (1, 2, 3)
   @test ∂(2) == (3, 2, 4)
   ∂ = Boundary{2,1}(t)
-  @test ∂(1) == (1, 3, 2)
-  @test ∂(2) == (1, 4, 5)
+  @test ∂(1) == (1, 2, 3)
+  @test ∂(2) == (2, 5, 4)
   ∂ = Boundary{1,0}(t)
-  @test ∂(1) == (3, 2)
-  @test ∂(2) == (1, 2)
+  @test ∂(1) == (1, 2)
+  @test ∂(2) == (2, 3)
   @test ∂(3) == (3, 1)
-  @test ∂(4) == (2, 4)
-  @test ∂(5) == (4, 3)
+  @test ∂(4) == (4, 3)
+  @test ∂(5) == (2, 4)
   𝒞 = Coboundary{0,1}(t)
-  @test 𝒞(1) == (2, 3)
-  @test 𝒞(2) == (4, 1, 2)
-  @test 𝒞(3) == (3, 1, 5)
-  @test 𝒞(4) == (5, 4)
+  @test 𝒞(1) == (1, 3)
+  @test 𝒞(2) == (5, 2, 1)
+  @test 𝒞(3) == (3, 2, 4)
+  @test 𝒞(4) == (4, 5)
   𝒞 = Coboundary{0,2}(t)
   @test 𝒞(1) == (1,)
   @test 𝒞(2) == (2, 1)
   @test 𝒞(3) == (1, 2)
   @test 𝒞(4) == (2,)
   𝒞 = Coboundary{1,2}(t)
-  @test 𝒞(1) == (2, 1)
-  @test 𝒞(2) == (1,)
+  @test 𝒞(1) == (1,)
+  @test 𝒞(2) == (1, 2)
   @test 𝒞(3) == (1,)
   @test 𝒞(4) == (2,)
   @test 𝒞(5) == (2,)
@@ -484,30 +484,30 @@ end
   ∂ = Boundary{2,0}(t)
   @test ∂(1) == (1, 2, 6, 5)
   @test ∂(2) == (6, 2, 4)
-  @test ∂(3) == (6, 4, 3, 5)
-  @test ∂(4) == (3, 1, 5)
+  @test ∂(3) == (5, 6, 4, 3)
+  @test ∂(4) == (1, 5, 3)
   ∂ = Boundary{2,1}(t)
-  @test ∂(1) == (1, 3, 5, 6)
-  @test ∂(2) == (3, 9, 4)
-  @test ∂(3) == (4, 7, 8, 5)
-  @test ∂(4) == (2, 6, 8)
+  @test ∂(1) == (1, 2, 3, 4)
+  @test ∂(2) == (2, 7, 8)
+  @test ∂(3) == (3, 8, 9, 5)
+  @test ∂(4) == (4, 5, 6)
   ∂ = Boundary{1,0}(t)
   @test ∂(1) == (1, 2)
-  @test ∂(2) == (3, 1)
-  @test ∂(3) == (6, 2)
-  @test ∂(4) == (4, 6)
-  @test ∂(5) == (5, 6)
-  @test ∂(6) == (1, 5)
-  @test ∂(7) == (4, 3)
-  @test ∂(8) == (3, 5)
-  @test ∂(9) == (2, 4)
+  @test ∂(2) == (2, 6)
+  @test ∂(3) == (6, 5)
+  @test ∂(4) == (5, 1)
+  @test ∂(5) == (5, 3)
+  @test ∂(6) == (3, 1)
+  @test ∂(7) == (2, 4)
+  @test ∂(8) == (4, 6)
+  @test ∂(9) == (4, 3)
   𝒞 = Coboundary{0,1}(t)
-  @test 𝒞(1) == (1, 6, 2)
-  @test 𝒞(2) == (9, 3, 1)
-  @test 𝒞(3) == (2, 8, 7)
-  @test 𝒞(4) == (7, 4, 9)
-  @test 𝒞(5) == (5, 8, 6)
-  @test 𝒞(6) == (3, 4, 5)
+  @test 𝒞(1) == (1, 4, 6)
+  @test 𝒞(2) == (7, 2, 1)
+  @test 𝒞(3) == (6, 5, 9)
+  @test 𝒞(4) == (9, 8, 7)
+  @test 𝒞(5) == (3, 5, 4)
+  @test 𝒞(6) == (2, 8, 3)
   𝒞 = Coboundary{0,2}(t)
   @test 𝒞(1) == (1, 4)
   @test 𝒞(2) == (2, 1)
@@ -517,14 +517,14 @@ end
   @test 𝒞(6) == (2, 3, 1)
   𝒞 = Coboundary{1,2}(t)
   @test 𝒞(1) == (1,)
-  @test 𝒞(2) == (4,)
-  @test 𝒞(3) == (2, 1)
-  @test 𝒞(4) == (2, 3)
-  @test 𝒞(5) == (3, 1)
-  @test 𝒞(6) == (4, 1)
-  @test 𝒞(7) == (3,)
-  @test 𝒞(8) == (3, 4)
-  @test 𝒞(9) == (2,)
+  @test 𝒞(2) == (1, 2)
+  @test 𝒞(3) == (1, 3)
+  @test 𝒞(4) == (1, 4)
+  @test 𝒞(5) == (4, 3)
+  @test 𝒞(6) == (4,)
+  @test 𝒞(7) == (2,)
+  @test 𝒞(8) == (2, 3)
+  @test 𝒞(9) == (3,)
   𝒜 = Adjacency{0}(t)
   @test 𝒜(1) == (2, 5, 3)
   @test 𝒜(2) == (4, 6, 1)
@@ -539,17 +539,17 @@ end
   𝒜 = Adjacency{2}(t)
   @test 𝒜(1) == (2, 3, 4)
   @test 𝒜(2) == (1, 3)
-  @test 𝒜(3) == (2, 4, 1)
+  @test 𝒜(3) == (1, 2, 4)
   @test 𝒜(4) == (1, 3)
 
   # 4 quadrangles in a grid
   elems = connect.([(1, 2, 5, 4), (2, 3, 6, 5), (4, 5, 8, 7), (5, 6, 9, 8)])
   t = HalfEdgeTopology(elems)
   𝒜 = Adjacency{2}(t)
-  @test 𝒜(1) == (3, 2)
+  @test 𝒜(1) == (2, 3)
   @test 𝒜(2) == (1, 4)
   @test 𝒜(3) == (1, 4)
-  @test 𝒜(4) == (3, 2)
+  @test 𝒜(4) == (2, 3)
 
   # invalid relations
   elems = connect.([(1, 2, 3), (4, 3, 2)])