Normally, we would swap two things or swap two things (them) around. "Swap around something" sounds odd to me.

"Swap" can be used in every case where you would "swap around" but the reverse is not true. "Swap" includes the idea of exchange, such as "I will swap my oranges for your apples". "Swap around" is more likely to be used with a meaning closer to "move around", such as "I swapped around the apples and oranges in my fruit bowl".

"Swap around"... the around is redundant is unnecessary, as "To Swap" implies movement.

To me, "swap around" suggests that things exchange places, but are still in the same larger area. "Swap" by itself is a little broader - you can swap (exchange) your things with another person, but that means your original possessions are not with you now. It can also mean that you take something, but leave something else.