Using refinement subdivision techniques, we construct smooth multiwavelet bases for L2(R) and L2([0,1]) which are in an appropriate sense dual to Alpert orthonormal multiwavelets. Our new multiwavelets allow one to easily give smooth reconstructions of a function purely from knowledge of its local moments. At the heart of our construction is the concept of moment-interpolating (MI) refinement schemes, which interpolate sequences from coarse scales to finer scales while preserving the underlying local moments on dyadic intervals. We show that MI schemes have smooth refinement limits. Our proof technique exhibits an intimate intertwining relation between MI schemes and Hermite schemes. This intertwining relation is then used to infer knowledge about moment-interpolating schemes from knowledge about Hermite schemes. Our MI multiwavelets make Riesz bases for L2 and unconditional bases of a variety of smoothness spaces, so they can efficiently represent smooth functions. We here derive an algorithm which rapidly develops a piecewise polynomial fit to data by recursive dyadic partitioning and then rapidly produces a smooth reconstruction with matching local moments on pieces of the partition. This avoids the blocking effect suffered by piecewise polynomial fitting.