Is selling a payer swaption = buying a receiver swaption?

I know the answer is No - because a receiver option is a call on bond prices (i.e. that interest rates will drop), and a payer option is a put on bond prices. Call and negative Put payoffs are not quite symmetric.

However, I know that selling a payer swaption means giving the couterparty the right to pay-fixed and recieve floating. Since I sold the payer swaption I am paying floating.

A receiver swaption is the right to receive fixed and pay floating.

Is the only difference the ownership of the exercise right? They seem like they’d offer similar hedging benefits.

Think of selling payer swaption vs selling receiver swaption + receiver swap?

The hedging beneits between selling a payer vs buying a reciever differ greatly. The first has negative convexity and the latter has positive convexity. Selling a payer option has unliminted loss if rates increase vs buying a receiver has limited loss of the premium if rates increase and unlimited gain(constrained by the level of rates) if rates decrease. As a result they will have differing deltas, differing durations and opposite convexity exposures.