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It has been proposed that machine learning techniques can benefit from symbolic representations and reasoning systems. We describe a method in which the two can be combined in a natural and direct way by use of hyperdimensional vectors and hyperdimensional computing. By using hashing neural networks to produce binary vector representations of images, we show how hyperdimensional vectors can be constructed such that vector-symbolic inference arises naturally out of their output. We design the Hyperdimensional Inference Layer (HIL) to facilitate this process and evaluate its performance compared to baseline hashing networks. In addition to this, we show that separate network outputs can directly be fused at the vector symbolic level within HILs to improve performance and robustness of the overall model. Furthermore, to the best of our knowledge, this is the first instance in which meaningful hyperdimensional representations of images are created on real data, while still maintaining hyperdimensionality.