Active thermal materials like thermal diodes, regulators, and switches have the potential to revolutionize thermal management, creating an opportunity for significant energy savings. We present results on a thermal switching composite that changes its thermal conductivity based on applied strain. The composite is constructed of highly-crystalline, high aspect ratio cellulose nanocrystal (CNC) nanorods embedded in a shape-memory polymer matrix. The properties of the matrix allow for changes to the thermal state to be indefinitely retained and also for the state to be reversed. A switching ratio of two is achieved for this proof-of-concept composite. By comparing the measured results to a Maxwell mixing model, the primary drivers of the thermal conductivity change are traced to changes in crystallinity of the matrix and CNC alignment.