Products of 2-Digit Pairs which Reversed reveal Same Product

Theorem
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:
 * $504, 756, 806, 1008, 1148, 1209, 1472, 1512, 2016, 2208, 2418, 2924, 3024, 4416$

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:

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:

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 $9$.