I know the community's favorite multi-dimensional infinite series of twisty puzzles are the cuboids(denoted lxmxn]), but I was wondering if anyone has attempted to catalog another such infinite series: the puckoids/cheesoids. I call these puzzles as such because the Rubik's Cheese and the Puck are the classic representatives of the series, though I am open to other naming suggestions.

The 'natural' shape for puzzles in this series would be a right circular cylinder. From here the cylinder would be divided laterally into n layers, and radially into m even wedges. Layers would turn in increments of 360/m degrees, and the m/2 wedges turn in increments of 180. As such, the Cheese is the 1-layer, 6-wedge member of the series, and the Puck is the 1-layer, 12-wedge member of the series. Looking to shape mods, the Rubik's UFO is 2-layer, 6-wedges, and the Masterball is 4-layers, 8-wedges.

Planar cuts would limit us to an even number of wedges, but if we allow non-planar cuts, Puckoids with an odd number of wedges can be conceptualized. if their a 2k+1 wedges, a move would divide the puzzle into k wedges one side an k + 1 wedges on the other.

Interestingly enough, there are some intersections between cuboids and puckoids:
1 wedge puckoids are mechanically equivalent to 1x1xn cuboids and are likewise trivial
2 wedge puckoids are mechanically equivalent to 1x2xn cuboids
4 wedge puckoids are mechanically equivalent to 2x2xn cuboids if rotations about the two equal axises are limited to 180 degrees.

In the following table, I will use the short hand nl, mw for n-layers, m-wedges. Puzzles that have been built(shape mods included) are highlighted in blue.

1l, 1w 1l, 2w 1l, 3w 1l, 4w 1l, 5w 1l, 6w 1l, 7w 1l, 8w 1l, 9w 1l, 10w 1l, 11w 1l, 12w
2l, 1w 2l, 2w 2l, 3w 2l, 4w 2l, 5w 2l, 6w 2l, 7w 2l, 8w 2l, 9w 2l, 10w 2l, 11w 2l, 12w
3l, 1w 3l, 2w 3l, 3w 3l, 4w 3l, 5w 3l, 6w 3l, 7w 3l, 8w 3l, 9w 3l, 10w 3l, 11w 3l, 12w
4l, 1w 4l, 2w 4l, 3w 4l, 4w 4l, 5w 4l, 6w 4l, 7w 4l, 8w 4l, 9w 4l, 10w 4l, 11w 4l, 12w
5l, 1w 5l, 2w 5l, 3w 5l, 4w 5l, 5w 5l, 6w 5l, 7w 5l, 8w 5l, 9w 5l, 10w 5l, 11w 5l, 12w
6l, 1w 6l, 2w 6l, 3w 6l, 4w 6l, 5w 6l, 6w 6l, 7w 6l, 8w 6l, 9w 6l, 10w 6l, 11w 6l, 12w

Please let me know if I have missed any.

You have to remember, though, that any odd nunber of wedges besides 1 would be impossible to turn without fudging, I belive.

Does the use of internal hollows to let pieces pass each other(as seen in the Kilominx and Oskar's Pentagon) qualify as fudging? Also, does the Pentagon fill the slot for 2l, 5w? One of Oskar's YouTube comments mentions that 90 degree "wedge turns" are possible, but that they "won't get you anywhere".

In my book the "Puckoids" are deep-cut prisms.
"Puckoids" share the axis system with the prisms just like the Dino Cube shares its axis system with the Master Skewb.
Take a look at the hexoid:
viewtopic.php?f=15&t=23424
It combines the piece sets of the Cheese with the hexagonal prism.

Besides that:
3l,4w is equivalent to the 2x2x3 and not highlighted blue.