This work describes a new variant of projective splitting for monotone inclusions, in which cocoercive operators can be processed with a single forward step per iteration. This result establishes a symmetry between projective splitting algorithms, the classical forward backward splitting method (FB), and Tseng's forward-backward-forward method (FBF). Another symmetry is that the new procedure allows for larger stepsizes for cocoercive operators: the stepsize bound is $2\beta$ for a $\beta$ cocoercive operator, which is the same as for FB. To complete the connection, we show that FB corresponds to an unattainable boundary case of the parameters in the new procedure. Unlike FB, the new method allows for a backtracking procedure when the cocoercivity constant is unknown. Proving convergence of the algorithm requires some departures from the usual proof framework for projective splitting. We close with some computational tests establishing competitive performance for the method.