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Im reading up on Polya counting, thanks for the mention.
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Polya counting is actually overkill, unless you are looking at much more complicated problems. This can be done with
Burnside's lemma. You might look at actions of S_4 on possible flops, and then for each conjugacy class of S_4, count how many flops are fixed by that symmetry. The identity fixes everything, 4-cycles fix nothing, 3-cycles fix flops that are monotone in the 4th suit or trips in the moving suits, etc. It's not too bad, but not easier than a direct computation here.