Transporting a mattress on its side is generally not a good idea. Mattresses are made to lie flat. So when they get bounced around in the back of a truck, they’re already under stress. Add being leaned on their side into the mix, and you’ve got a recipe for warped mattresses and damaged internal support cores.
This is especially true for mattresses with springs. Pure foam mattresses might handle gentle transportation over short distances on their sides if you have absolutely no other option. But any time on its side can damage the spring system in a hybrid or coil mattress. Protecting your mattress’s internal structure is also why you should avoid folding it during transport.