I like the idea of a swivel, but it's another point of failure, and a potentially nasty one. On balance, I'd opt for not using one. (But someone may come up with a good reason why a swivel is essential, in which case I'll change my mind!)
For wire rope the hardware you use on the end really should be matched to the rope. For instance, using a swivel on some rotation resistant rope can reduce the rope's strength 30% or more.
I'd check with your rope supplier to make sure they are properly matched.
For synthetic, I have no idea if it matters or not.