This entire year I fished two different setups for swimjigs with the same reel and line, the only difference being the rod. Both rods are 7' and the action fast for both. The MH rod is rated for 3/8-1oz while the Medium is rated for 1/4-5/8oz.
Now most videos I watch from tour pros to avid anglers usually are heavy or medium-heavy recommendations for swimjigs; articles as well. Previously I've always went with the industry standard...MH-fast. But, I seem to prefer a more pronounced parabolic bend with medium rods as I'll throw a lot of squarebills and bladed jigs. I must admit that the caveat here is that I do prefer lighter wire swimjigs to allow me to fish lighter line and typically don't fish heavy gauge 5/0 swimjigs.
What I observed for the entire year fishing both setups equally is that I lost less fish on the medium setup. I've attribute this to the parabolic action of the medium rod as it seemed ideal if they did break surface that there was less power whipping the rod back with slack; rod loaded better on hookups. I had no problems with hookups on either setup due to the lighter wire hooks; always made sure they were sharp.
I did have more line breakages running lighter line on the MH setup, but expected it in running at times when I should've cut back some line and retied. Although, this laziness was catered to with the medium rod which gave me no issues with breaking off on hook sets. Likewise, I was able to sweep a little more on hooksets with the medium rod. With the MH I really had to reel set which can be ideal when swimjigging and an ideal technique for some; MH apparently the more efficient solution for reel setting.
Lastly, casting distance of course goes to the medium rod as it loads better, especially when throwing qtr oz swimjigs. Accuracy was also better for me personally with the medium rod.
Of course, they make medium heavy and heavy action rods with moderate fast actions to cater to my preferences and will look to try a few out. Anyway, feel free to share your preferences as I'm curious to know if others prefer lighter power rods for swimjigging.