Convolutional Neural Networks (CNNs) are the predominant model used for a variety of medical image analysis tasks. At inference time, these models are computationally intensive, especially with volumetric data.In principle, it is possible to trade accuracy for computational efficiency by manipulating the rescaling factor in the downsample and upsample layers of CNN architectures.However, properly exploring the accuracy-efficiency trade-off is prohibitively expensive with existing models.To address this, we introduce Scale-Space HyperNetworks (SSHN), a method that learns a spectrum of CNNs with varying internal rescaling factors.A single SSHN characterizes an entire Pareto accuracy-efficiency curve of models that match, and occasionally surpass, the outcomes of training many separate networks with fixed rescaling factors.We demonstrate the proposed approach in several medical image analysis applications, comparing SSHN against strategies with both fixed and dynamic rescaling factors.We find that SSHN consistently provides a better accuracy-efficiency trade-off at a fraction of the training cost. Trained SSHNs enable the user to quickly choose a rescaling factor that appropriately balances accuracy and computational efficiency for their particular needs at inference.