This classification of quadrilaterals is almost, but not quite, the same as their classification by symmetry. The only exception is that the generic trapezoid has no symmetry, like the generic quadrilateral.
I worked out the classification of all N-gons by symmetry only a few weeks ago. The group of the regular polygon is the dihedral group D2n of order 2n. If m times i is 2n, then m is the order of either just 3 subgroups (if m and i are both even), or just 1 (if either is odd), up to conjugacy.
The subgroups are Dm if i odd
Cm if m odd
Dm-, Dm+, Cm if m and i are even,
where the reflections of Dm+ permute the vertices evenly, while those of Dm- permute them oddly. (alternatively, one type of reflection fixes a vertex, the other an edge).
Every group except Cn is the group of some type of polygon.
For n = 4 we have D8 square
D4+ C4 D- rect'l (none) rhombus
D2+ C2 D2- iso-trap para kite
Sorry - those corresponding arrays were cramped a bit too closely.
For n = 6 we get
D6+ C6 D6-
D2- C2 D2+
for the groups. D12 - regular
D6+ edges same length, vertices form two equilat triangles of distinct sizes
D6- edges alternate in length, alternate sides produced would give two equilat triangles of distinct sizes
C6 can't happen
C3 vertices form two equilat triangles of difft sizes (but same center) in most general way