We investigate the effects of spin-orbit interaction (SOI) on the exchange cotunneling through a spinful Coulomb blockaded quantum dot. In the case of zero magnetic field, Kondo effect is shown to take place via a Kramers doublet and the SOI will merely affect the Kondo temperature. In contrast, we find that the breaking of time-reversal symmetry in a finite field has a marked influence on the effective Anderson and Kondo models for a single level. The nonlinear conductance can now be asymmetric in bias voltage and may depend strongly on the direction of the magnetic field. A measurement of the angle dependence of finite-field cotunneling spectroscopy thus provides valuable information about orbital and spin degrees of freedom and their mutual coupling.