On your dual mag, do you have 2 P-leads on the left side(shower of sparks) and one on the right? Or when you pull the prop through in the normal direction do you hear a loud clunk like a deadbolt lock(Impulse coupling)? My guess is you have a shower of sparks and that kind of limits your troubleshooting options. From your symptoms I would bet that your left starting points(shower of sparks) are working and your left mag points are not set correctly. When you turn your key to energize your starter, it grounds both LH and RH points and only the LH start points operate. When it kicks and you go to both then both the LH and RH points operate as normal. This is so the normal points don't fire the spark plugs during start at 20 degrees BTDC and kick back. If you have a impulse coupling your left mag is most likely bad and your starting on your right.
To prove the Mag/start switch is working correctly: Caution you are moving the prop with the mags HOT if there is fuel in the cylinder the prop can swing through and hurt you.
1) You can test your ignition switch by putting a Mag timing buzz box on the P leads(if studs) mag switch to both, move the prop to get both buzz box lights to come one. Then sit in the pilot seat, master off, wiggle the switch in the off position to see if the buzz box light change, then right and the right light should change, wiggle the key, then go to left, wiggle the key, move fully clock wise to the start circuit(mast off) and wiggle again. If you have P-lead caps you can remove both normal caps and connect the buzz box clips to the p-leads and run the same test above.