In Thailand, when they want to rejuvinate a large mango tree, they pretty much cut it back to almost nothing but trunk. They may leave some of the larger branches but even they are cut waaaaay back. It looks odd and your first thought is "what the hell?!". But everyone does it over there and that includes the big fruit parks.

Here's an article I found that may yield some info. I could not access it here from work.
www.ehow.com/how_4524185_prune-mango-tree.html

Here's some shots from Thailand when I was there last. This is at Suphattraland, a big fruit park. You can see in the pic the mangos that have been pruned and those that have not.

Here's another shot. These are not pruned back as severely as some I've seen. I've seen some reduced to not much more than a trunk.

Might be best to wait until the one is done blooming/fruiting. If the other is not going to bloom, you might start with that one. Another thing to consider if you cut these back is to obtain some cuttings from some really good cultivars and graft onto the ends of some of the branches you cut back. Might especially be a good idea for the one half of the mango that doesn't bloom. Good luck.

If you are an area that gets a lot of wind, the flowers could have blown off. This is what happens to some of our mango trees. Again, you could prune it 'way back'!
To post a photo, you need to post it on a URL such as www.photobucket.com. Then you pick the option 'HTML Code' and copy & post it where you want it to appear.