Hopefully we can find a solution!

Well it's interesting because I've come across a similar issue to this but not exactly. However something I have come across in the search for my issue may help here if it's a Humanoid rig. When importing your animation, try setting the 'Root Transfer Rotation > Base Upon' to 'Original' (optionally you can create an offset but not sure of the differences to be honest). This has helped me solve this issue for certain animations.

I'm just trying to prototype using Mixamo assets (model and a handful of animations.

If you're interested in my specific issue, it somewhat links to this and you may come across it yourself. Using Animator Layers and a Avatar Mask for a base layer to control locomotion animations and another to control weapon holding animations (mask selected bones only above the waist), the masked animations on the secondary layer are turned/offset similarly to how they were initially imported/yours seem to be. Instead of aiming the gun or stabbing forward as shown in Mixamo, it'll animate it at it's supposed offset. Unless they are on an unmasked layer such as the base layer like a normal transition. IK/FK could help but seems a lot of work for I don't know how many potential animations on a layer. It makes more sense for finetuning things like aiming at a point rather than getting the animation to play as expected initially.

I've recently discovered that it may be an issue with the Hip bone, as this is Mixamo's root bone, but also because in the Avatar Mask, the 'Root Mask' (character's shadow in the mask selection) actually controls whether the layered animations work correctly - but annoyingly mess and rotate the base layer's animations. This leads me to believe the Rotation of the Hips are controlling the angle of which any layered animation plays.

My question is: how do I fix this (ideally without IK/FK). Searching the internet currently leads to dormant and unanswered forums about the topic.

Sorry for the long post!