Spins and oscillators are foundational to much of physics and applied sciences. For quantum information, a spin 1/2 exemplifies the most basic unit, a qubit. High angular momentum spins (HAMSs) and harmonic oscillators provide multilevel manifolds which have the potential for hardware-efficient protected encodings of quantum information and simulation of many-body quantum systems. In this work, we demonstrate a new quantum control protocol that conceptually merges these disparate hardware platforms. Namely, we show how to modify a harmonic oscillator on demand to implement a continuous range of generators to accomplish linear and nonlinear HAMS dynamics. The spinlike dynamics are verified by demonstration of linear spin coherent [SU(2)] rotations, nonlinear spin rotations, and comparison to other manifolds like simply truncated oscillators. Our scheme allows universal control of a spin cat logical qubit encoding with interpretable drive pulses: We use linear operations to accomplish four logical gates and further show that nonlinear spin rotations can complete the logical gate set. Our results show how motion on a closed Hilbert space can be useful for quantum information processing and opens the door to superconducting circuit simulations of higher angular momentum quantum magnetism.