Equal edges:
(4,n,2n) + (3,4,4,n)
Infinite family, works for any n, including spherical cases n = 3, 4, 5 (and Euclidean n = 6, but that is par for the course).
Corresponds to diminishing operation on cantellated regular tiling (3,4,n,4).
{n,n/2} + {3,n}
Infinite family corresponding to diminishing of {3,n} by replacing vertices with n-gons. Both types of vertex allow for mixing under the rule [n] <-> [3,3]; for example, for n = 7 (first hyperbolic case), the allowed vertices are (3,3,3,3,3,3,3), (3,3,3,3,3,7), (3,3,3,7,7), and (3,7,7,7).
(3,4,8,40) + (3,5,8,8)
(3,5,6,18) + (3,6,6,9)
(3,3,10,10) + (4,4,5,5)
Allows mixed vertex (3,4,5,10). Hybrid tilings known.
(3,5,12,12) + (4,5,5,12)
Hybrid tilings known.
(4,4,10,10) + (5,5,6,6)
Allows mixed vertex (4,5,6,10). Hybrid tilings known.
{5,4} group
Contains vertex (3,4,10,20), vertex (3,3,3,3,4,4), and mixed vertex (3,3,4,5,5) ([3,3,4] <-> [5,5])
{18,4} group
Contains vertex (3,3,4,4,6,6) and mixed vertex (3,4,6,18,18) ([3,4,6] <-> [18,18])
{4,5} group
Contains vertex (4,4,10,30).
{10,5}/{5,6}/{3,10} group
{10,5} is a part of the infinite family {n,n/2}, so it automatically meshes with {3,10} and mixed vertices: (3,3,10,10,10,10), (3,3,3,3,10,10,10), (3,3,3,3,3,3,10,10), and (3,3,3,3,3,3,3,3,10).
However, the same edge length is also shared by {5,6}, which allows for three more mixed vertices: (3,3,3,3,3,5,5,5), (3,3,3,5,5,5,10), and (3,5,5,5,10,10).
In addition, the edge length of this group is double of that of the {5,4} group, allowing for mixing with polygons from there.
(3,4,4,5,5,5) group
There is a mixing rule [4,5] <-> [3,20]. Through its application, we get vertices (3,3,4,5,5,20) and (3,3,3,5,20,20).
{4,7} group
There is a mixing rule [4,4] <-> [3,7]. This gives us vertices (3,4,4,4,4,4,7), (3,3,4,4,4,7,7), and (3,3,3,4,7,7,7)
{8,6}/{4,8} group
Allows for a mixed vertex (4,4,4,4,8,8,8).
{30,6}/{5,15/2} group
While {5,15/2} doesn't have an actual legal vertex, it still allows for mixed vertex (5,5,5,5,5,30,30).
(5,5,5,5,5,6,6,6,6,6) group
There is a mixing rule [5,6] <-> [4,12]. We get vertices (4,5,5,5,5,6,6,6,6,12), (4,4,5,5,5,6,6,6,12,12), (4,4,4,5,5,6,6,12,12,12), (4,4,4,4,5,6,12,12,12,12), and (4,4,4,4,4,12,12,12,12,12)
{5,12}/{12,10} group
Allows for a mixed vertex (5,5,5,5,5,5,12,12,12,12,12).
(4,12)^5/(5,6)^5 group
Allows for 4 types of mixed vertices.
(6,10)^10/(4,5)^12 group
Allows for 1 mixed vertex (6,10)^5+(4,5)^6.
(3,12n)^20n/(12n,20n)^15n family
Mixing rule is [3,3,3,3,12n] <-> [20n,20n,20n], leading to (5n-1) mixed vertices in-between.
For example, for the smallest group, (3,12)^20/(12,20)^15, the mixed vertices are:
(3^16,12^19,20^3)
(3^18,12^18,20^6)
(3^8,12^17,20^9)
(3^4,12^16,20^12)
Random results (most likely just coincidences that fit within the stated precision):
5, 5, 10, 10, 13, 17 + 5, 6, 6, 12, 17, 18 (1:1): 2.421014466950248
4, 7, 8, 12, 19, 19 + 5, 6, 7, 10, 12, 17 (1:1): 2.4210618605909233
7, 10, 17, 18, 20, 20 + 11, 11, 11, 12, 13, 16 (1:1): 2.5536322668571807
(4,6,7,9,9,9,11,11,11)+(3,3,4,7,7,7,7,11,11,11)
(3,5,5,7,7,7,11,11,12,12)+(3,6,6,6,6,7,9,10,10,12)
(3,4,6,6,7,8,9,11,11,12)+(4,5,5,5,5,5,10,11,12,12)
Double edges (first vertex is the one with doubled edges):
(n,n,2n,2n,2n,2n) + (4,4,2n,2n)
Infinite family. Case n = 5 is special because the smaller edge is the "(4,4,10,10) + (5,5,6,6)" case.
{12,4} + (3,3,12,12)
{8,8} + {8,4}/{3,8}
{6,9} + {9,4}
{3,18}/{18,9} + {18,4} and its group
{5,10} + {10,4}
{4,12} + {12,4}
{3,14}/{14,7} + {7,4}
{9,18} + (4,4,18,18,18,18)
Random results:
(3,5,9,10,10) + (8,8,8,9,9,11,12,12,12,12)