I've done some research online about how to blow out my sprinkler lines and everything I've read says not to push air through the backflow preventer. However, I have no other place to hook up an air line other than before the backflow preventer. Does this mean that I have a special case, or does it mean mine was installed incorrectly?
I don't know why blowing through a backflow preventer would be a problem. Although the newer ones have spring activation. Yours is relatively old ( I have the same one) and the bronze check plate simply drops open if there is no water pressure to hold it closed . Although I doubt if it will reseat if you shut off water pressure to connect the air . I suspect air pressure will not reseat it ( unless there is a very large volume of high pressure air.) and you will not be able to blow into the lines. Why do you want to blow lines ? Normally to prevent ice cracking pipes one opens a drain(s) at low points . It is installed correctly.
Could you detach the green hose? You could blow from there.
Backflow valve is a a wheel. Water doesn't flow quickly, so it is ok. But with air, you may get much higher spinning, which may break the wheel. On the other hand, backflow valves are at risk of freezing (small volumes, special geometries, so freezing could damage it).
So, if you can detach the green hose, this would be ideal. Else you should try to blow water gently. Ev. close all but one opening (at a time), so that air will flow slower.