I posted about this earlier. There is, as far as I'm concerned, a bug in the game concerning tire changes on some cars (including the Lancia). If you buy certain tires and DON'T install them at the time of purchase, they won't show up in the tire list for that car on the tuning screen, despite the fact that the store claims you own one or more sets of them.
The solution is to purchase a set of snow tires again, and this time install them when it asks right after you buy them. I had the same problem with the Soft Sport tires on the Lancia.
If this isn't a bug, I have no idea what the logic behind it is.