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Jelly, (削除) 43 41 40 (削除ここまで) 18 bytes

3RŒḄIṙ
4RÇ€
OIṠ¢e@

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Or all test cases

###How?

An infinity word has:

1. the same first and last letter;
2. length 5;
3. no equal letters next to each other;
4. sum of its four alphabet deltas equal to zero;
5. sum of its four alphabet deltas signs equal to zero;
6. two positive alphabet deltas or two negative alphabet deltas in a row.

This can all be boiled down to a condition that the alphabet delta signs are some rotation of [1,1,-1,-1] (where the sign of 0 is 0)

###How?

3RŒḄIṙ - Link 1, make a rotation of [1,1,-1,-1]: n
3R - range(3) -> [1,2,3]
 ŒḄ - bounce -> [1,2,3,2,1]
 I - increments -> [1,1,-1,-1]
 ṙ - rotate left by n (e.g. n=1 -> [1,-1,-1,1])
4RÇ€ - Link 2, make the four rotations of [1,1,-1,-1] (no arguments, AKA niladic)
4R - range(4) -> [1,2,3,4]
 Ç€ - call the last link as a monad for €ach
OIṠ¢e@ - Main link: word
O - cast to ordinals
 I - increments - the alphabet deltas
 Ṡ - sign
 e@ - exists in (@ = reversed arguments)
 ¢ - call last link as a nilad