You can keep the cut 120 backing for 127 cameras, if you control what part of the 120 you keep. If you cut to keep the edge that has the 6x4.5 track,
and leave it spooled backward, the film you create will work (with a little extra space between frames) in 4x4 cameras. The same cut and spooling will work with 4x6 cm and 4x3 cm, using the 120 6x6 track. This latter, requires masking the film gate in the camera by a small amount; I've got a 127 Baby Ikonta 4x3 that works great with this cut, using about a 3mm strip of black paper taped on one edge of the film gate (supply side, so the film pulls it flat rather than crumpling it).
You get a longer strip than original 127; you'll get 12 exposures on 4x6 (would have been 8), 16 exposures (instead of 12) on 4x4, and 24 on 4x3 (would have been 16 originally). And yes, the roll will still fit on a 127 spool; it fills up to the very edge of the flange, instead of leaving a little flange beyond the backing paper as with original 127.
There's an Italian 3D seller who 3D prints
a cutter for this that can be used in daylight, including respooling back to original direction -- which only works for 4x3 and 4x6 formats -- same who makes a 126 cartridge for use with 35 mm film and no backing