tiles = Select[
Subsets[Select[Subsets[Range@6, {2}],
UnsameQ @@ Ceiling[#/2] &], {3}], Sort@Flatten@# == Range@6 &];
tileGraph[tile_, face_, rotation_] :=
Module[{edgePoints =
RotateLeft[
Flatten[If[
OrderedQ@#, {2 # - 1, 2 #} &@
Position[PolyhedronData["Tetrahedron", "EdgeIndices"], #][[
1, 1]], {2 #, 2 # - 1} &@
Position[PolyhedronData["Tetrahedron", "EdgeIndices"],
Reverse@#][[1, 1]]] & /@
Partition[PolyhedronData["Tetrahedron", "FaceIndices"][[face]],