As Dave noted, Padilla throws a ton of fastballs. He is fourth among starters behind sinker ball specialists Aaron Cook, Mike Pelfrey and Rick Porcello, in fastball percentage at 75%. Fastballs, on a pitch-by-pitch basis, show a large platoon split. So a pitcher with lots of fastballs should show an more extreme split. Curveballs and changeups generally show little platoon split. Padilla throws a splitter/changeup and a curveball, but he does so rarely and they are poor (changeup/spiltter) or average (curveball) pitches.
Padilla throws both a two-seam fastball and four-seam fastball. To RHBs, he throws the four-seam 35% of the time and two-seam 45% of the time, so to RHBs he throws fastballs 80% of the time. To lefties, he goes 55% four-seam and 16% two-seam. This makes sense: of fastballs, two-seamers have a larger platoon split, so it would be better to throw them more to RHBs and limit them against LHBs.
I plotted the location of all of this fastballs (green four-seam and blue two-seam) to LHBs and RHBs this year. The figures are messy, but should give a general idea of the location he goes with the two pitches.
To righties, you can see a kind of diagonal orientation of his two-seam fastballs running from up-and-in through the center of the zone to down-and-away. This is good: within the zone, pitches do best up-and-in and down-and-away. His two-seam to RHBs is thrown regularly to these locations. With his four-seam he mostly stays away.
To LHBs, the orientation of his two-seam would be opposite of ideal, where hitters do better. Because of this, he does not throw it as often and there is no general trend to where it ends up. He is left throwing his four-seam fastball much more, concentrating it on the outside half.
The differences are seen in the value of the pitches. Both of the fastballs are good to RHBs (0.4 runs above average for his four-seam and 0.7 for his two-seam), but against LHBs things fall apart (0.8 runs below average for the the four-seam and a horrid 2.5 runs below average for the two-seam).
As Dave noted, this is a really questionable call against a team with some good lefties.