An important issue to understand when working with FBX, is Blender always points the bone primary axis in a +Y direction, while Maya and FBX support unique bone orientation per-bone. You can set joint orientations in the Maya "Orient Joint Options" dialog.
Maya Joint Orientation in a Minute - YouTube
The Blender FBX Import setting "automatic bone orientation" is a one-way fix that tries to reorient the bones in the parent->child direction, but this does not always work, and creates a new skeleton which does not match the original FBX skeleton.
IMO, when using Blender in an FBX import/export workflow with other tools and game engines, "automatic bone orientation" should always be turned off, because it merely hides problems in bone orientations. With this setting off, Blender should no longer break FBX rigs and animations when importing. If the bones are pointing in the wrong directions, you can turn on the bone-axis display, and you will see which primary axis each bone was authored for.
The two solutions I'm aware of to have Blender FBX rigs and animations to interoperate with other tools are...
To standardize on a skeleton with all bones pointing in +Y direction. One way to do this is to create the original skeleton/rig in blender, and export it to other tools to use for work.
If you have an existing skeleton built elsewhere with non-uniform bone orientation, you can import just the skeleton into Blender, use Automatic bone-orientation and/or editing to reorient the bones, then export this new "+Y bones" skeleton into your game engine as a new Skeleton/Avatar. You can use this skeleton to build animations in Blender, export them onto that Skeleton in the game engine, and then use the game-engine retargetting tools to convert them to the other skeleton with non-uniform bone orientations.
For more, see this bug...
⚓ T53620 FBX Import - Armatures are incorrect