No, permanent licenses are not cross-platform.
What Adobe has done is to let you have a Mac license without paying another $700, but you don't get to keep the old one.
That's why they call it a swap.
If you do want a Mac/PC license, you could go with the $10 monthly Photography Program. There you can install a Mac and a PC version for your two activations. You get upgrades forever, but like your light bill if you decide to quit (and it is a one year commitment) paying, Photoshop quits working.
Gene