Canonical Form of Rotated and Color Swapped Rubik's Cube

So I was given an interview question, and I am extremely confused by their response. I'm not going to give away their details (if you know, you know), but the problem is interesting.

Let's say you have a standard Rubik's cube with White facing Up, Green facing Front, Red facing Right, and Yellow, Blue, and Orange opposite those colors in the same order (I.E. Blue opposite Green, Red - Orange, White - Yellow).

Say you wanted to write a database that contained solutions to the every possible state the cube could be in (solved state, top rotated once, top rotated twice, etc.). I understand that there are 24 unique ways you could rotate this cube to bring it back into a canonical form to avoid all possible cases.

Now let's say you allow non-standard colors (I.E. Red could be opposite White, Blue opposite Yellow, and on and on).

My understanding is that permuting the colors of the cube amongst the six posible colors is kinda of a one way function and there is no way to really undo it.

Hence, if you change whatever cube you are given with any possible color scheme into one where you replace whatever color is up with White, the front with Green, and so on, you essentially solve the problem of finding a canonical form.

The other engineers contest that it is possible and that you should permute all the colors first, and then try to rotate it into a canonical form from there.

Am I missing something, or is this like a trick question problem to befuddle engineers?

I tried reaching out to them but they won't get back to me with the "solution".