Quantum computation presents a powerful new paradigm for information processing. A robust universal quantum computer can be realized with any well controlled quantum system, but a successful platform will ultimately require the combination of highly coherent, error-correctable quantum elements with at least one entangling operation between them(1,2). Quantum information stored in a continuous-variable system-for example, a harmonic oscillator-can take advantage of hardware-efficient quantum error correction protocols that encode information in the large available Hilbert space of each element(3-5). However, such encoded states typically have no controllable direct couplings, making deterministic entangling operations between them particularly challenging. Here we develop an efficient implementation of the exponential-SWAP operation(6) and present its experimental realization between bosonic qubits stored in two superconducting microwave cavities. This engineered operation is analogous to the exchange interaction between discrete spin systems, but acts within any encoded subspace of the continuous-variable modes. Based on a control rotation, the operation produces a coherent superposition of identity and SWAP operations between arbitrary states of two harmonic oscillator modes and can be used to enact a deterministic entangling gate within quantum error correction codes. These results provide a valuable building block for universal quantum computation using bosonic modes.