Find the Infinity Words! (Note: This is a spin-off of my previous challenge Find the Swirling Words!)

Definition of Infinity Word:


If you connect with curves all the characters of an Infinity Word on the alphabet (A-Z) you obtain the infinity symbol &infin; like in the diagrams below.
All the even connection must be down, all the odd connections must be up.
You can ignore upper/lowercase or consider/convert all to upper case or all to lower case.
The input words are only characters in the alphabet range of A-Z, no spaces, no punctuation, or symbols.
Each word must be exactly 5 characters. Words > 5 or &lt; 5 are not valid.
If a word has double consecutive characters, the word is not valid, like "FLOOD" or "QUEEN".
All the Infinity Words start and end with the same character.


Here there are some examples:



Task:

Write a full program or function that will take a word from standard input and will output if is an Infinity Word or not. The output can be true/false, 1/0, 1/Null, etc.

Test cases:

Infinity Words:
ALPHA, EAGLE, HARSH, NINON, PINUP, RULER, THEFT, WIDOW

NOT Infinity Words:
CUBIC, ERASE, FLUFF, LABEL, MODEM, RADAR, RIVER, SWISS, TRUST, 
KNEES, QUEEN, GROOVE, ONLY, CHARACTER, OFF, IT, ORTHO


Rules:


Shortest code wins.


Optional Task:

Find, as a list, as many Infinity Words as you can in an English dictionary. You can take for example as reference the complete list of English words here.
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)