like...

|55|44|66|
|44|66|55|

so figure out how a piece can move.

pick any piece, try to move it somewhere.
when you move a piece you have to move it's pair at the same time.
when you move to a piece it's pair has to move at the same time too.
a piece always becomes a pair with the piece it moves to.
no matter how many pairs, there's only one answer to how a piece can
move.

A common problem, I forget what it's called.

There's only one answer for how any piece can move.

A piece always goes where a piece leaves.

No piece can move to where a piece moves back where it came from.

No such thing as a free space, a piece always moves to another piece.

A pair never moves to a pair.

so try this...

draw for each piece a line from one piece to another that connects
each piece to move from the first piece until the last piece that goes
back where it starts.

see this as a machine diagram.

move a piece then figure the machine diagram again, it's the same
machine.