I think spinning reels are called such because they spin the line rather like an old spinning jenny.
First thing to check. When you load the line, it must come off the spool you bought in the same direction as the reel is winding it on. Most reels (not old Mitchells) wind in a clockwise direction, looking from the front. To counter this the line must come off in an anit-clockwise direction from the spool as you look down onto it.
Some spools aren't designed with this in mind, they're square or have cut-outs or lips. Also the spools you buy should be near enough the same on diameter as the spool you load on to. Difficult since most have a bigger diameter, but that's the best way to eliminate spin when loading the line.
Your reel will spin the line with each cast and retrieve. Try wrapping some tape or decorative ribbon onto your spool so that it's all flat and tie a big weight to the end. Open the bail and pull the tape off as if you were casting and wind in again. Do this half a dozen times if you can and see the twist you have put into the tape.
Some lines are better than other at taking the spin. Braids are the worst, especially the soft flimsy ones. Swivels don't improve things much as far as spinning the line is concerned, it's all an old wives tale.
