I went to the shop and measured my 39 bumpers and they sure seem identical to me. I could not see a difference, maybe Gene will see this and add to the answer.
My 1938 bumpers have a very slight difference in the curves between front and rear. What clued me in as I bought a rear splash pan from Gene and it only matched the curve of one of my bumpers. I recall Gene verifying this years back but not sure what model years were being discussed besides my 38.
They are so close they can be bolted front or rear with some persuasion but the splash pan only fits the rear bumper mounted on the rear in my case.
The rear bumper bars is the same for 1938 and 1939 cars with no expoded spare tires. The front bumper bars are the same for both years EXCEPT the front bar in 1939 had more bow or curve.
Looks like Chevrolet superseded the front 39 service bumper 350646 by the rear bumper 597785 in 1940. Both the 41 and 42 catalog shows 597785 as the front and rear bumpers for the 39 JA.
So it looks like the rear will work on the front but not the other way around.
The 1939 book has 350646 for the front and 597785 for the rear. They must have expected you to form the bar to the brackets to make the front curve. In 1948 all original front and rear bumpers were discontinued for 1937-1940 cars and small trucks and they sold a universal bumper. It was a 1941 version with lots of extra holes to match the various bracket arangements.