Find all integer pairs that produce a given Loeschian number
05AB1E, 11 9 bytes
Thanks to @ovs for -2!
ÝãʒãÀ¦POQ

Oh my, I am getting in the hang of code golfing!
Prints a list of all the valid pairs (e.g [[1, 0], [0, 1]]). If there are none, the list is empty ([]). Also outputs both of any reverse integer pairs.
Try it online!
How?
You may count this as a port of the other answers, but I took a look only at the Husk answer before writing the program!
Ý             # Push a list of all numbers from 0 to the input.
 ã            # Push the cartesian power of lists. (Basically, finding all possible pairs)
  ʒ           # For each pair...
   ãÀ¦        # Find all other permutations of the pair.
      P       # Multiply each permutation.
       O      # Add the products.
        Q     # If the result is not equal to the input, yeet (throw) them from the list.
              # Automatically print the pairs not yeeted.