You’re correct that receiver swaptions will be exercised when interest rates are low and payer swaptions will be exercised when interest rates are high. In that sense, they’re similar to call options and put options (respectively) on bonds.
S2000 to expand a little further to the question… When calculating the duration of a swap.
The variable side is multiplied by .5 of the pmt period. For the Fixed side I’ve seen a few times where the Fixed Duration is multiplied by .75? Is the .75 part of the formula or does something trigger it?