Jelly, (削除) 43 41 40 25 24 23 22 21 14 (削除ここまで) 13 bytes

-7 bytes thanks to fireflame241 (0ị=1ị$->=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

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###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.

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)

Jelly, (削除) 43 41 40 25 24 23 22 21 14 (削除ここまで) 13 bytes

-7 bytes thanks to fireflame241 (0ị=1ị$->=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 "RULES")

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.

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)

Jelly, (削除) 43 41 40 25 24 23 22 21 (削除ここまで) 14 bytes

-7 bytes thanks to fireflame241 (0ị=1ị$->=ṚḢ and use of IIA=2,2 to test for the 4 rotations)

=ṚḢȧOIṠIIA=2,2

TryItOnline
Or all test cases (plus "RULES")

###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.

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=2,2 - 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)
 2,2 - literal [2,2]
 = - equals? (non-vectorising version)

Jelly, (削除) 43 41 40 25 24 23 22 21 14 (削除ここまで) 13 bytes

-7 bytes thanks to fireflame241 (0ị=1ị$->=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 "RULES")

###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.

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)

Jelly, (削除) 43 41 40 25 24 23 22 (削除ここまで) 21 bytes

9BḤ’ṙ4R¤
0ị=1ị$aOIṠe¢

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###How?

9BḤ’ṙ4R¤ - Link 1, make the four rotations of [1,1,-1,-1] (no arguments, AKA niladic)
9B - 9 converted to binary  -> [ 1, 0, 0, 1]
 Ḥ - double (vectorises) -> [ 2, 0, 0, 2]
  ’ - decrement (vectorises) -> [ 1,-1,-1, 1]
 ¤ - nilad followed by link(s) as a nilad
 4R - range(4) -> [ 1,  2, 3, 4]
 ṙ  - rotate left by -> [[-1,-1,1,1],[-1,1,1,-1],[1,1,-1,-1],[1,-1,-1,1]]
0ị=1ị$aOIṠe¢ - Main link: word
 $  - last two links as a monad
0ị - element at index 0 (last character)
 =  - equals?
 1ị - element at index 1 (first character)
 a  - and
  O - cast word to ordinals
 I - increments - the alphabet deltas
 Ṡ - sign (vectorises)
 e - exists in
 ¢ - call last link (1) as a nilad

Jelly, (削除) 43 41 40 25 24 23 22 21 (削除ここまで) 14 bytes

-7 bytes thanks to fireflame241 (0ị=1ị$->=ṚḢ and use of IIA=2,2 to test for the 4 rotations)

=ṚḢȧOIṠIIA=2,2

TryItOnline
Or all test cases (plus "RULES")

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=2,2 - 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)
 2,2 - literal [2,2]
 = - equals? (non-vectorising version)