Great! At least I’m right on track on my understanding.
In summary, payer (pay fix rate) swaption is similar to call option on interest rates and put option on fixed income securities because as interest rates rises your option increase in value because:
a) Payer Swaption: current swap rate increases and you benefit from receiving a higher floating rate than fix
b) Call Option on Interest Rates: Interest rates increase, you receive the value due to differential rates
c) Put Option on Fixed Income: Interest rates increase, fixed income securities price drops and a put option increases in value.