We construct new examples of derived autoequivalences, for a family of higher-dimensional Calabi-Yau varieties. Specifically, we define endo- functors of the bounded derived categories of coherent sheaves associated to varieties arising as the total spaces of certain natural vector bundles over complex Grassmannians. These functors are defined using Fourier- Mukai techniques, and naturally generalize the Seidel-Thomas spherical twist for analogous bundles over complex projective spaces. We prove that they are autoequivalences. We also give a discussion of the motivation for this construction, which comes from homological mirror symmetry.