This expression is unusual in that, although it seems clear that it is a jokey variant of ‘ass-backwards’, it actually predates that phrase by a good fifty years.
It is certainly American in origin – the word ‘ass’ isn’t used to mean rear in most other English-speaking countries.
It may be that the explicit use of ‘ass’ in ‘ass-backwards’ was the reason that the coy ‘bass-ackwards’ was initially preferred.
Oddly, ‘ass-backwards’, although it alludes to things being wrong and muddled, is (the last time I looked) the usual positioning. It shares that strangeness with ‘head over heels‘, which is also the normal way of things.
The earliest example that I can find in print is from the Kansas newspaper the Eureka Sentinel, July 1878:
A traveling gymnast gave a public exhibition on Main St. last Thursday evening. His contortions and double somersaults “bass backwards” was hard to beat.
The earliest example that I can find of ‘ass-backwards’ is 1934.