There are two basic footwork patterns that you can do, and they're both easy. They both take 2 beats, so they can be done on the 1+2 of a swingout, or repeated for switches.
Are you ready? for the big Reveal? In no particular order, the two footwork patterns are:
Step-Step: step on alternate feet evenly on every beat (1 2)
Kick-Ball-Change*: kick with your right foot, step on your right foot, step on your left foot (1+2)
These two footwork patterns are interchangeable because they take the same number of beats and you wind up on the same foot at the end. The Kick-Ball-Change is just a gussied up version of the Step-Step.
Let's take a look at these in action - my apologies for repeating clips, but I want to use ones where you can see their feet.
Step-Step Swivels (I call these "Jewel swivels" because this is her standard footwork)
Buck Privates at 1:49 - Jewel just steps on every beat, starting by sitting deep into the 7+8 of the previous swingout. On the even beats (8, 2, 4, 6, etc.) she steps to the left with her left foot, and then brings her right foot to meet it on the odd beats (1, 3, 5, 7), all while maintaining her turnout.
Kick-Ball-Change Swivels (I call these "Jean swivels" -- guess why?)
Groovie Movie at 3:45 - it's hard to see if you aren't looking for it, but Jean is doing KBC swivels. You can tell because her toe comes up slightly on 1, which doesn't happen if you're putting your weight down. It's not a big kick from the knee, it's just a little flick. Try keeping your heel slightly touching the ground.
Groovie Movie at 5:36 - here you can see Irene Thomas (brunette in the center) and Kay Smith doing KBC swivels alongside Jean. It's nice to see some other LA follows featured in switches besides just Jean and Jewel!
The footwork pattern you choose is completely personal choice, and it's not dictated by the lead at all. Use the music as your cue - I think that Step-Step swivels are a little smoother, and KBC swivels are snappier. You can travel cover more distance with Step-Step swivels than KBC, and when the music's really fast you won't have enough time between the beats for Kick-Ball-Changes, so Step-Steps are your only option. Practice both and play with changing them up!
*Kick-Ball-Change = KICK/step on the BALL of same foot/CHANGE your weight to opposite foot
It's not "Kick the old ball and chain"...