Hi @zaradin, I don’t have a feel for what those metric values mean; can you share some screenshots of the registration quality? I.e. overlay reoriented__registered_image.nii on top of MNI152_T1_1mm.nii, and adjust the display settings on the former to show the alignment quality.
The L/R<->R/L flip should not cause issues - flirt and other FSL tools are usually agnostic w.r.t. handedness.