Exchange bias is commonly manifested as the hysteresis-loop shift observed when a ferromagnet is in contact with an antiferromagnet. Here, we report observations of exchange bias with unusual features of a ferromagnet in contact with a spin glass, demonstrating that this is a phenomenon of greater generality. The easily measured properties of the ferromagnet allow access to the internal magnetic degrees of freedom of the glass to which they are coupled. Our results show that a Co/CuMn bilayer system exhibits all the rich phenomena of coercivity enhancement, bias-field shifts and training effects associated with a conventional ferromagnet/antiferromagnet system. Nevertheless, striking differences arise, such as an orientation reversal of the bias field in a small temperature range just below the blocking temperature. We argue that all features can be understood within the context of a random-field model for long-ranged oscillatory Ruderman-Kittel-Kasuya-Yosida (RKKY) coupled spins.