Earlier quoted context omitted.
I tried it on my computer. Converting from JPEG to JPEG XL and then back to JPEG gets you a bit-wise identical JPEG file.
I'm guessing the opposite (XL -> JPEG -> XL) does not work, i.e., "losslessly convert back and forth with regular JPEG".
1. native JPEG XL lossy compression
2. JPEG lossy compression, but using a more space-efficient storage format than classic JPEG
3. lossless compression
The lossless conversion between JPEG XL and JPEG only works with 2).
Normally you'd only use 2) for existing JPEG images in order to avoid a lossy re-compression, and use 1) for any freshly-created images because native JPEG XL compression will compress the image data even more efficiently than the JPEG compatibility-mode, though if you've got strong interoperability constraints with clients requiring JPEG data you might possibly want to store even newly created images using sub-format 2) in order to get the ability to losslessly convert back and forth between JPEG XL and JPEG.