Theorem: Products of 2-Digit Pairs which Reversed reveal Same Product

The following positive integers can be expressed as the product of $2$ two-digit numbers in $2$ ways such that the factors in one of those pairs is the reversal of each of the factors in the other:
:$504, 756, 806, 1008, 1148, 1209, 1472, 1512, 2016, 2208, 2418, 2924, 3024, 4416$
<!-- fascists won't include the damn thing {{OEIS|A289978}}  SIEG HEIL! SIEG HEIL!-->

Proof:
Let $n \in \Z_{>0}$ such that:
:$n = \sqbrk {a b} \times \sqbrk {c d} = \sqbrk {b a} \times \sqbrk {d c}$
where $\sqbrk {a b}$ denotes the two-digit positive integer:
: $10 a + b$ for $0 \le a, b \le 9$
from the Basis Representation Theorem.
We have:
{{begin-eqn}}
{{eqn | l = \paren {10 a + b} \paren {10 c + d}
      | r = \paren {10 b + a} \paren {10 d + c}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 100 a c + 10 \paren {a d + b c} + b d
      | r = 100 b d + 10 \paren {b c + a d} + a c
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 99 a c 
      | r = 99 b d
      | c = 
}}
{{eqn | ll= \leadsto
      | l = a c 
      | r = b d
      | c = 
}}
{{end-eqn}}
Thus the problem boils down to finding all the sets of one-digit integers $\set {a, b, c, d}$ such that $a c = b d$, and so that:
:$n = \sqbrk {a b} \times \sqbrk {c d} = \sqbrk {b a} \times \sqbrk {d c}$
and also:
:$n = \sqbrk {a d} \times \sqbrk {b c} = \sqbrk {d a} \times \sqbrk {c b}$
Thus we investigate all integers whose divisor count is $3$ or more, and find all those which have the product of single-digit integers in $2$ ways, as follows:
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} 4
      | r = 3
      | c = {{DCFLink|4}}
}}
{{eqn | ll= \leadsto
      | l = 4
      | r = 1 \times 4
      | c = 
}}
{{eqn | r = 2 \times 2
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 12 \times 42
      | r = 21 \times 24
      | c = 
}}
{{eqn | r = 504
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} 6
      | r = 4
      | c = {{DCFLink|6}}
}}
{{eqn | ll= \leadsto
      | l = 6
      | r = 1 \times 6
      | c = 
}}
{{eqn | r = 2 \times 3
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 12 \times 63
      | r = 21 \times 36
      | c = 
}}
{{eqn | r = 756
      | c = 
}}
{{eqn | l = 13 \times 62
      | r = 31 \times 26
      | c = 
}}
{{eqn | r = 806
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} 8
      | r = 4
      | c = {{DCFLink|8}}
}}
{{eqn | ll= \leadsto
      | l = 8
      | r = 1 \times 8
      | c = 
}}
{{eqn | r = 2 \times 4
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 12 \times 84
      | r = 21 \times 48
      | c = 
}}
{{eqn | r = 1008
      | c = 
}}
{{eqn | l = 14 \times 82
      | r = 41 \times 28
      | c = 
}}
{{eqn | r = 1148
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} 9
      | r = 3
      | c = {{DCFLink|9}}
}}
{{eqn | ll= \leadsto
      | l = 9
      | r = 1 \times 9
      | c = 
}}
{{eqn | r = 3 \times 3
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 13 \times 93
      | r = 31 \times 39
      | c = 
}}
{{eqn | r = 1209
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {10}
      | r = 4
      | c = {{DCFLink|10}}
}}
{{eqn | ll= \leadsto
      | l = 10
      | r = 1 \times 10
      | c = and so does not lead to a solution
}}
{{eqn | r = 2 \times 5
      | c = 
}}
{{end-eqn}}
Further integers $n$ such that $\map {\sigma_0} n \le 4$ need not be investigated, as one of the pairs of factors will be greater than $9$.
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {12}
      | r = 6
      | c = {{DCFLink|12}}
}}
{{eqn | ll= \leadsto
      | l = 12
      | r = 1 \times 12
      | c = which does not lead to a solution
}}
{{eqn | r = 2 \times 6
      | c = 
}}
{{eqn | r = 3 \times 4
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 23 \times 64
      | r = 32 \times 46
      | c = 
}}
{{eqn | r = 1472
      | c = 
}}
{{eqn | l = 24 \times 63
      | r = 42 \times 36
      | c = 
}}
{{eqn | r = 1512
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {16}
      | r = 5
      | c = {{DCFLink|16}}
}}
{{eqn | ll= \leadsto
      | l = 16
      | r = 1 \times 16
      | c = which does not lead to a solution
}}
{{eqn | r = 2 \times 8
      | c = 
}}
{{eqn | r = 4 \times 4
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 24 \times 84
      | r = 42 \times 48
      | c = 
}}
{{eqn | r = 2016
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {18}
      | r = 6
      | c = {{DCFLink|18}}
}}
{{eqn | ll= \leadsto
      | l = 18
      | r = 1 \times 18
      | c = which does not lead to a solution
}}
{{eqn | r = 2 \times 9
      | c = 
}}
{{eqn | r = 3 \times 6
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 23 \times 96
      | r = 32 \times 69
      | c = 
}}
{{eqn | r = 2208
      | c = 
}}
{{eqn | l = 26 \times 93
      | r = 62 \times 39
      | c = 
}}
{{eqn | r = 2418
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {20}
      | r = 6
      | c = {{DCFLink|20}}
}}
{{eqn | ll= \leadsto
      | l = 20
      | r = 1 \times 20
      | c = which does not lead to a solution
}}
{{eqn | r = 2 \times 10
      | c = which does not lead to a solution
}}
{{eqn | r = 4 \times 5
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {24}
      | r = 8
      | c = {{DCFLink|24}}
}}
{{eqn | ll= \leadsto
      | l = 24
      | r = 1 \times 24
      | c = which does not lead to a solution
}}
{{eqn | r = 2 \times 12
      | c = which does not lead to a solution
}}
{{eqn | r = 3 \times 8
      | c = 
}}
{{eqn | r = 4 \times 6
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 34 \times 86
      | r = 43 \times 68
      | c = 
}}
{{eqn | r = 2924
      | c = 
}}
{{eqn | l = 36 \times 84
      | r = 63 \times 48
      | c = 
}}
{{eqn | r = 3024
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {36}
      | r = 9
      | c = {{DCFLink|36}}
}}
{{eqn | ll= \leadsto
      | l = 36
      | r = 1 \times 36
      | c = which does not lead to a solution
}}
{{eqn | r = 2 \times 18
      | c = which does not lead to a solution
}}
{{eqn | r = 3 \times 12
      | c = which does not lead to a solution
}}
{{eqn | r = 4 \times 9
      | c = 
}}
{{eqn | r = 6 \times 6
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 46 \times 96
      | r = 64 \times 69
      | c = 
}}
{{eqn | r = 4416
      | c = 
}}
{{end-eqn}}
{{ProofWanted|The challenge remains to prove, without going through all cases exhaustively, that there are no further pairs.}}
{{Improve|Why not write out the upper triangular part of the multiplication table for $1 \le n \le 9$, $n \ne 5, 7$, and find the common numbers in the table? </br> $5, 7$ can be eliminated via a simple argument $p x \ge 10$, and there are only $28$ numbers left to compare}}