  We develop a collisional formalism adapted for the dynamics of ultracold
dipolar particles in a confined geometry and in fields tilted relative to the
confinement axis. Using tesseral harmonics instead of the usual spherical
harmonics to expand the scattering wavefunction, we recover a good quantum
number $\xi = \pm 1$ which is conserved during the collision. We derive the
general expression of the dipole-dipole interaction in this convenient basis
set as a function of the polar and azimuthal angles of the fields. We apply the
formalism to the collision of fermionic and bosonic polar KRb molecules in a
tilted electric field and in a one-dimensional optical lattice. The presence of
a tilted field drastically changes the magnitude of the reactive and inelastic
rates as well as the inelastic threshold properties at vanishing collision
energies. Setting an appropriate strength of the confinement for the fermionic
system, we show that the ultracold particles can even further reduce their
kinetic energy by inelastic excitation to higher states of the confinement
trap.
