The problem of interpolating a rigid body motion is to find a spatial trajectory between a prescribed initial and terminal pose. Two variants of this interpolation problem are addressed. The first is to find a solution that satisfies initial conditions on the k-1 derivatives of the rigid body twist. This is called the kth-order initial value trajectory interpolation problem (k-IV-TIP). The second is to find a solution that satisfies conditions on the rigid body twist and its k-1 derivatives at the initial and terminal pose. This is called the kth-order boundary value trajectory interpolation problem (k-BV-TIP). Solutions to the k-IV-TIP for k=1,...,4, i.e. the initial twist and up to the 4th time derivative are prescribed. Further, a solution to the 1-IV-TBP is presented, i.e. the initial and terminal twist are prescribed. The latter is a novel cubic interpolation between two spatial configurations with given initial and terminal twist. This interpolation is automatically identical to the minimum acceleration curve when the twists are set to zero. The general approach to derive higher-order solutions is presented. Numerical results are shown for two examples.