  The shape and orientation of dark matter (DM) halos are sensitive to the
micro-physics of the DM particle, yet in many mass models, the symmetry axes of
the Milky Way's DM halo are often assumed to be aligned with the symmetry axes
of the stellar disk. This is well-motivated for the inner DM halo but not for
the outer halo. We use zoomed cosmological-baryonic simulations from the Latte
suite of FIRE-2 Milky Way-mass galaxies to explore the evolution of the DM
halo's orientation with radius and time, with or without a major merger with a
Large Magellanic Cloud (LMC) analog, and when varying the DM model. In three of
the four CDM halos we examine, the orientation of the halo minor axis diverges
from the stellar disk vector by more than 20 degrees beyond about 30
galactocentric kpc, reaching a maximum of 30--90 degrees depending on the
individual halo's formation history. In identical simulations using a model of
self-interacting DM with $\sigma = 1 \, \mathrm{cm}^2 \, \mathrm{g}^{-1}$, the
halo remains aligned with the stellar disk out to $\sim$200--400 kpc.
Interactions with massive satellites ($M \gtrsim 4 \times 10^{10} \,
\rm{M_\odot}$ at pericenter; $M \gtrsim 3.3 \times 10^{10} \, \rm{M_\odot}$ at
infall) affect the orientation of the halo significantly, aligning the halo's
major axis with the satellite galaxy from the disk to the virial radius. The
relative orientation of the halo and disk beyond 30 kpc is a potential
diagnostic of SIDM if the effects of massive satellites can be accounted for.
