Find the Infinity Words!
Jelly, 43 41 40 25 24 23 22 21 14 13 bytes
-7 bytes thanks to fireflame241 (0ị=1ị$-&gt;=ṚḢ and use of IIA⁼2,2 to test for the 4 rotations)
-1 Thanks to Kevin Cruijssen (use of previously unavailable nilad Ø2 which yields [2,2])
=ṚḢȧOIṠIIA⁼Ø2

TryItOnline
Or all test cases (plus &quot;RULES&quot;)
How?
An infinity word has:

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

All but (1) and (equivalently) (4) may 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)
fireflame241 noted that this is then equivalent to the deltas of the deltas of the alphabet delta signs being in [[2,2],[2,-2],[-2,2],[-2,-2]] which may be tested by the absolute values being equal to [2,2]!
How?
=ṚḢȧOIṠIIA⁼Ø2 - Main link: word
 Ṛ            - reverse word
=             - equals? (vectorises)
  Ḣ           - head (is the first character equal to the last?)
   ȧ          - and
    O         - cast word to ordinals
     I        - increments - the alphabet deltas (or just [] if 1st != last)
      Ṡ       - sign (vectorises)
       I      - increments - deltas of those signs
        I     - increments - deltas of those
         A    - absolute value (vectorises)
           Ø2 - literal [2,2]
          ⁼   - equals? (non-vectorising version)