Technology ownership and knowledge transfer aim at the dissemination of public research results, usually in one direction - from the university or public research organisation (PRO) that produced the results. However, the complementarities between these types of instruments are not clear compared to the bidirectional channels of knowledge exchange. We analyse one PRO in Spain, a country that is peculiar in terms of infrequent changes to the legislation on science and patents and a predominance of PRO-owned over PRO-invented patents, similar to the situation in the USA but different from that in most EU member states. Against this background, knowledge transfer (measured through PRO-invented patents) increases faster than technology ownership (measured through PRO-owned patents). This situation may be damaging to knowledge exchange (measured through PRO co-owned patents).