“Pull-down”, to use your preferred terminology, means taking something progressive and making it frame-interlaced. The term can actually be applied to 50Hz material (25fps stored as 50i), or to 60Hz material if it’s 30fps progressive stored as 60i: you can talk about “2:2 pulldown” as opposed to “3:2 pulldown”. In the 2:2 case you’ve got each progressive frame split into a pair of interlaced fields. In the 60Hz 3:2 case, as you say, not only is each frame split into fields, but some fields are repeated in an alternating sequence.
To see what happens in an ideal world, check out my 1080i/60 wedge pattern (which is called 1080i_60_deinterlace_test.mkv - I think it’s actually the same pattern as one @Scottosan 's [pulldown/FHD/00712.m2ts] but repackaged as mkv). This a 1080p/24fps video stored as 1080i/60 (so 3:2). It’s VC-1: leave deinterlacing set to Auto-select. When it plays, my Vero 4K+ not only recognises that the video is frame-interlaced, it also actually switches output mode to 1080p/24Hz.
But switching the output mode like this is obviously not something you can do unless the entire video is consistent; if it switches from frame- to field-interlaced on the fly, or if it switches from 3:2 to 2:2 on the fly (less common) then, during the frame-interlaced 3:2 sections, probably the best you can hope for is to reconstruct the original frames and output those in a 3:2:3:2 pattern with 60Hz output. (And if you’re dealing with 480i/60 there isn’t even a 480p/24Hz mode to switch to).
You do, however, need to make sure you don’t output any frames that consist of fields from two different original frames woven together. The pattern of repeated fields looks like this:
Frame 1, odd field
Frame 1, even field
Frame 2, odd field
Frame 2, even field
Frame 2, odd field again
Frame 3, even field
Frame 3, odd field
Frame 4, even field
Frame 4, odd field
Frame 4 even field again.
So, if this is misdiagnosed as 2:2 instead of 3:2, then you will end up with five frames instead of four, and the third and fourth output frame in each set of 5 will consist of fields from two different frames combined.
So, for 60Hz material, it’s not enough to determine if it’s frame-interlaced or field-interlaced; if it’s frame-interlaced you also need to check if it’s 3:2 or 2:2, because that affects which fields are assembled into how many frames.
In terms of the expected output, if it’s 60Hz 2:2 you would probably expect either 30Hz output or 60Hz output with each frame repeated twice. If it’s 60Hz 3:2 you would expect 60Hz output with frames repeated in a 3:2 pattern, so:
Frame 1
Frame 1
Frame 2
Frame 2
Frame 2
Frame 3
Frame 3
Frame 4
Frame 4
Frame 4.
Or, if the video is frame-interlaced and 3:2 all the way through, switching the output mode to 24Hz should work correctly!
The exception to this, of course, is if you have 120Hz output. Then you can actually switch between 30fps with each frame shown 4 times, 24fps with each frame shown 5 times or 60fps generated from field-interlaced 60i with each frame shown twice, and do all that on the fly, without changing output mode. But plenty of TVs don’t support 120Hz input, so that’s not something to worry about as a first pass.