Recent experiments report that slowly sheared noncolloidal particle suspensions unexpectedly exhibit rate(omega)-dependent complex viscosities in oscillatory shear, despite a constant relative viscosity in steady shear. Using a minimal hydrodynamic model, we show that van der Waals attraction gives rise to this behavior. At volume fractions phi = 20-50%, the complex viscosities in both experiments and simulations display power-law reductions in shear, with a phi-dependent exponent maximum at phi = 40%, resulting from the interplay between hydrodynamic, collision, and adhesive interactions. Furthermore, this rate dependence is accompanied by diverging particle diffusivities and pronounced cluster formations after repeated oscillations.  Previous studies established that suspensions transition from reversible absorbing states to irreversible diffusing states when the oscillation amplitude exceeds a ϕ-dependent critical value γc0,ϕ. Here, we show that a second transition to irreversibility occurs below an ω-dependent critical amplitude, γc0,ω≤γc0,ϕ, in the presence of weak attractions.