There is much controversy in the literature over the relationship between the openness of firms' innovation strategies and firm characteristics such as size, RD intensity and sector. We argue that the controversy arises because, both theoretically and empirically, only a binary, open vs. closed, strategy has been considered. In this paper, we distinguish among three firm strategies: open, semi-open and closed, drawing upon a panel of Spanish firms (2004-2006) using data from Community Innovation Survey (CIS)-type surveys, and two different indicators of openness. Our results show that open innovators are smaller and less RD intensive than semi-open ones, although larger and more RD intensive than closed innovators. These results reduce some of the controversies, and show that two conflicting forces, absorptive capacity and a oneedo effect, are at stake in open innovation strategies.