I've suggested on the wiki classifying patterns by their symmetry. This seems to be now well underway.

However, a lot of oscillators cannot be placed in the usual planar symmetry classes, since they have glide symmetry some sort (along the time axis). How should these symmetry classes be denoted? I think this list covers all the possibilities:

Glide rotations

—any single phase is asymmetric, sum of all phases has S2 symmetry (blocker)

—any single phase is asymmetric, sum of all phases has S4 symmetry (dinner table)

—any single phase has S2 symmetry, sum of all phases has S4 symmetry (windmill)

Glide reflections

—any single phase is asymmetric, sum of all phases has D1 symmetry (orthogonally: griddle; diagonally: mold) (glide reflectiv spaceships also fall here!)

—any single phase has S2 symmetry, sum of all phases has D2 symmetry (orthogonally: Achim's p144, diagonally: bipole)

— any single phase has S4 symmetry, sum of all phases has D4 symmetry (phoenix, Achim's p16)

Glide rotations+reflections

—any single phase has D1 symmetry, sum of all phases has D2 symmetry (orthogonally: smiley, tumbler; diagonally: mazing)

—any single phase has D2 symmetry, sum of all phases has D4 symmetry (orthogonally: blinker, monogram; diagonally: washing machine)

These are of course a subset of the three-dimensional line groups, but the Sn/Dn notation apparently does not extend to them.

There's also a question of what to do with billiard tables that require induction coils. A tubber stabilized by two bookends is D2; stabilized by four snakes it can be S4; stabilized by four houses, D4. A similar case is emulators, which can be either "cis" (falling then in the same class as smiley), or "trans" (falling then in the same class as Achim's p144).