I would think about a receiver swap first. You receive, so you sold a stream of little libors to your counterparty. Also assume a level move first instead of a steepener, etc...
For simplicity, think of the curve in 2 pieces ShortEnd, LongEnd. A steepener can occur if ShortEnd stays the same and the LongEnd goes up. It can also occur if the ShortEnd goes down and LongEnd stays the same. These cases will have different effects on your swaption.
Obviously, in the real world, the curve can have much more complicated moves.