All 3 digits must have the same parity, so the odd digits can be analyzed separately from the even digits.

The mod4 value of a 3-digit number is the same as the mod4 value of the last 2 digits.  

Evens: Consider flipping the last 2 digits.  What pairs of even digits have the same mod4 value when reversed?  26, 48 (or 04, but zero is excluded).  The 2 digits must differ by 4, so the only triplet would be {0,4,8} but zero is excluded.  So there is no solution with even digits.

Odds:  if both digits of a 2-digit number are odd, which ones do not have their mod4 value changed by reversing the digits?  15, 19, 37, 59  Again, the difference must be 4 (or a multiple of 4).

The only triplet which can be formed from these pairs is:  {1,5,9}.

Confirmed with a short program.

('1', '5', '9')

159 3

195 3

519 3

591 3

915 3

951 3

------------

from itertools import combinations

from itertools import permutations

for comb in combinations('123456789', 3):

    remainder = int(''.join(comb)) % 4

    winner = True

    for perm in permutations(comb):

        x = int(''.join(perm))

        if x % 4 != remainder:

            winner = False

            break

    if winner:

        print(comb)

        for perm in permutations(comb):

            x = int(''.join(perm))

            print(x, x%4)