Fusing satellite imagery acquired with different sensors has been a long-standing challenge of Earth observation, particularly across different modalities such as optical and Synthetic Aperture Radar (SAR) images. Here, we explore the joint analysis of imagery from different sensors in the light of representation learning: we propose to learn a joint, sensor-invariant embedding (feature representation) within a deep neural network. Our application problem is the monitoring of lake ice on Alpine lakes. To reach the temporal resolution requirement of the Swiss Global Climate Observing System (GCOS) office, we combine three image sources: Sentinel-1 SAR (S1-SAR), Terra MODIS and Suomi-NPP VIIRS. The large gaps between the optical and SAR domains and between the sensor resolutions make this a challenging instance of the sensor fusion problem. Our approach can be classified as a feature-level fusion that is learnt in a data-driven manner. The proposed network architecture has separate encoding branches for each image sensor, which feed into a single latent embedding. I.e., a common feature representation shared by all inputs, such that subsequent processing steps deliver comparable output irrespective of which sort of input image was used. By fusing satellite data, we map lake ice at a temporal resolution of <1.5 days. The network produces spatially explicit lake ice maps with pixel-wise accuracies >91.3% (respectively, mIoU scores >60.7%) and generalises well across different lakes and winters. Moreover, it sets a new state-of-the-art for determining the important ice-on and ice-off dates for the target lakes, in many cases meeting the GCOS requirement.