Neutron star (NS) mergers where both stars have negligible spins are commonly considered as the most likely 'standard' case. In globular clusters, however, the majority of NSs have been spun up to millisecond (ms) periods and, based on observed systems, we estimate that a non-negligible fraction of all double NS mergers ($\sim 4\pm 2\, {{\ \rm per\ cent}}$) contains one component with a spin of a (few) ms. We use the Lagrangian numerical relativity code SPHINCS_BSSN to simulate mergers where one star has no spin and the other has a dimensionless spin parameter of chi = 0.5. Such mergers exhibit several distinct signatures compared to irrotational cases. They form only one, very pronounced spiral arm and they dynamically eject an order of magnitude more mass of unshocked material at the original, very low electron fraction. One can therefore expect particularly bright, red kilonovae. Overall, the spinning case collisions are substantially less violent and they eject smaller amounts of shock-generated semirelativistic material. Therefore, the ejecta produce a weaker blue/ultraviolet kilonova precursor signal, but - since the total amount is larger - brighter kilonova afterglows months after the merger. The spinning cases also have significantly more fallback accretion and thus could power late-time X-ray flares. Since the post-merger remnant loses energy and angular momentum significantly less efficiently to gravitational waves, such systems can delay a potential collapse to a black hole and are therefore candidates for merger-triggered gamma-ray bursts with longer emission time-scales.