A direct modification of the Miles SB16 driver has some advantages over the swapstereo patch:
1) It doesn't require an unofficial build of DOSBox; and if you're using an unofficial build that doesn't have the swapstereo patch, it will work there as well.
2) It targets only the SB audio. The swapstereo patch reverses the channels of all inputs to the mixer.
3) It has an effect at the point where it's needed. The swapstereo patch is unaware of when the channels first become reversed, which can be a problem when the channels are reversed in-game, but not in videos.
A brief explanation of the issue with the driver: it sets up two samples worth of data for its initial stereo DMA transfer, but for some reason instructs the DSP to use only one sample, resulting in reversed channels because DOSBox keeps track of "leftover" samples from previous transfers. The issue with DOSBox is avoided by changing the driver to use both samples; and the modified driver still works correctly on a real SB16.
To modify the driver, run the attached program inside DOSBox in a folder where SB16.DIG is located (preferably an installed copy). The driver file has several variations, so the program searches for the correct offset to modify. If you encounter a version of SB16.DIG that the program does not recognize, and you're sure it is a Miles AIL3 driver, send me a PM so I can examine it.