How to canonicalize UndirectedEdge with good performance?

Question

When programming with graphs, it is sometimes important that UndirectedEdge[a,b] and UndirectedEdge[b,a] are treated as equivalent. Up to M12.0 it was possible to handle this by temporarily setting Orderless on UndirectedEdge. But in M12.1, UndirectedEdge can have three arguments, so Orderless cannot be used.

What is a good alternative that does not compromise peformance?

This is the workaround I have been using in IGraph/M:

canonicalEdgeBlock::usage = "canonicalEdgeBlock[expr] evaluates expression while making sure that all UndirectedEdge expressions inside are ordered canonically.";
SetAttributes[canonicalEdgeBlock, HoldAllComplete]

(* In M12.1 and later, UndirectedEdge can have 3 arguments, so we cannot canonicalize simply with Orderless. *)
(* TODO  The workaround /; Not@OrderedQ[{a, b}] is 10x slower than Orderless! *)
If[$VersionNumber >= 12.1,
  canonicalEdgeBlock[expr_] :=
      Internal`InheritedBlock[{UndirectedEdge},
        Unprotect[UndirectedEdge];
        UndirectedEdge[a_, b_, rest___] /; Not@OrderedQ[{a, b}] := UndirectedEdge[b, a, rest];
        expr
      ]
  ,
  canonicalEdgeBlock[expr_] :=
      Internal`InheritedBlock[{UndirectedEdge},
        SetAttributes[UndirectedEdge, Orderless];
        expr
      ]
]

It comes at a cost of a 5x to 10x slowdown. I am looking for solutions that are faster.

Here's an example use of this utility function in IGraph/M.