Can I move the opening of my screen door from the middle to the left?
Your post says "screen door" but it sounds like you want to change the sliding glass door and the screen along with it. Most glass door units are modular, the glass sections (stationary and sliding) are separate pieces and the track runs the whole width of the unit. If this is the case it may be possible to switch things around.

Take a good look at the unit, the sliding door should lift up and inwards for removal. Once removed you may be able to see whether or not the stationary units can be removed. Look for screws on the inside and outside that are holding the units in place. The stationary units can usually be lifted up and removed as well. If we represent the original arrangement as A-B-C (with "B" being the sliding door) you might be able to replace them in this configuration:
 B-C-A. 

The panels are heavy, don't try it alone. Prepare a flat area to lay down the sliding door so you can refurbish/lubricate the wheels while you have it off (I use sawhorses). Some old glass units are NOT tempered safety glass and could hurt you terribly if broken, be careful. The weatherstripping scheme, as well as the latch, may require some ingenuity as the setup will not be the same.

Not all glass door assemblies are made to take apart like described, especially custom stuff.