Here I tried to recapture the dent on Indy's right side in Raiders of the Lost Ark. The one on the left was the closest I could duplicate it on the normal hat.  The one on the 90 degree hat on the right, comes much closer, and pretty much forms itself.    The little white line on the hat is some hot glue used to attatch some ribbon or decoration, which was missing when I bought it, and why I got it for $12.50 in the women's dept. at Sears. It demonstrates that it's at 90 degrees.
90 degree turn creates a Raiders specific shape
What was the front of the hat is now over your right ear. The hat used should have an oval, not round hole, or turning the hat will have no effect.  So far this is all more theory than accepted fact, so don't do anything rash.
The images below on this side are of a hat in the normal position.
The images below on this side are of the same hat turned 90 degrees.
    It's not too aparent here, but the brim of the 90 degree hat mimicks the asymetry of the Raiders hat brim in every way, while the normal hat brim is completely symetrical. The hat  has a short brim with a folded and sewn edge, wherein there is a nylon filament which gives the soft wool brim the ablility to behave like a snap brim.  This filament limits as well as exagerates the brim's movements. The high, flat radius located at the front of Indy's right  sideburn from which the brim starts to angle down, is impossible to acheive with the filament in place, but the brim is trying to do it. Notice that the 90 degree hat puts a more extreme downward tilt to the front of the brim, a Raiders fedora trademark.   The unbashed crown on this hat is only 5 inches, which makes the curves happen a little more suddenly than on a true Indy hat.                                                                                             The thing the hat is on is a cement sculpture I once made of the Raiders hat.  The end that's inside the hat has a sharp right angle corner which puts a  real lump at the back of the hat above the crown/brim junction- a curved scull in there wouldn't make such a lump, and a hatband and ribbon would probably smooth things out a bit too .                                       I did'nt want to make this page without a proper Indy hat to begin with. The hat I have here can't even be worn without pushing out the top bash because the crown is so short. It's 100% wool and totaly soft, so it probably distorts too easily. I guess I'll have to order one of the real things now. It'll probably take a while to get here, so this is the best I can do for now. 
note: This hat originaly had some material wraped around the middle of the crown to give it an hourglass shape. (woman's hat)  leaving permanent dents. Otherwise the sides of the crown directly below would be even straighter. 
    The famous Raiders dimensional cut brim - It seemed it didn't exist before gear collectors started doing it themselves - then talking fedora manufacturers into doing it for them.  The black hat used on this page is at the left. It looks like the maker took a round  piece of felt and formed an oval hole in it, therefore the brim on the sides is wider (about 1/4 inch) than the brim is in the front and back. If this hat is worn turned 90 degees, you instantly get the famous Raiders Dimensional Cut Brim. I propose that the Herbert Johnson worn in Raiders of the Lost Ark was a standard fedora worn sideways.
There are 3 pictures on this page,  from the film, of the left side of the hat bulging out bigtime. That's not happening as much on the right side.- If the oval in Indy' hat is egg shaped, like the Peters bros. hat, (above left), and the hat is turned so that the front of the egg shaped oval  is over the right ear, there will be a greater volume, over a large area over the left ear, where the bulge is most prominent.  The bulge on the right side should be smaller sharper, and more centralized.  I think it materializes in the monkey on shoulder picture, forming a dent just above the ribbon just above the right ear.
NORMAL
90 degrees
When Indy's hat isn't being stretched front  to back by his scull, is the oval trying to return to it's original sideways orientation? In the dock scene, the hat is resting lightly on  top of his head.
    The back of the brim of Indy's hat in Raiders is at an angle. The hat at the left is the 90 degree hat,  naturaly assuming this angle. The hat on the right is  turned 180 Degees the other way, and the angle slants in the opposite direction. Now the normal front of the hat is over the left ear instead of the right. (I have atificially stretched these photos.)
  Here I bashed the hat at 45 degrees, and the top gets this ying-yang shape. The front bashes don't look good to me and the brim becomes only marginally tilted in back.
   I kind of just noticed this, I wasn't looking for it.  I knew about the bump on the back of the right bump of the crown, but I thought it was dents made by Indy's right hand fingers, pushing the hat down on his head. BUT - I'm getting the same little bump with the 90 degree turn. It doesn't happen with the normal crease, and surprisingly, not with the 180 degree from the 90 crease. I realize shadows are being cast in the flying wing picture, but the left hand bump of Indy's hat is fairly featureless in comparison, and the left and right bumps look the same as they do here in the Peruvian temple, Cairo, ect..for instance, when he is checking the booby trap floor stone in the temple.. (note: The top bashes have been creased, leaving lines left to right across the crown.) ...I'm starting to believe this theory!
90 degrees
Normal
180 away from 90 (backwards of 90)