Conflicting opinions on rough volatility motivate us to propose a convex combination of the rough Heston (rough 1/2) and smooth 3/2 models to create a novel 4/2 rough and smooth (4/2RS) volatility model. This parsimonious two-factor model captures many stylized facts from empirical studies and flexibly provides realistic variance distributions and rich autocorrelation structures. For instance, it generates an elasticity of variance (EV) of the variance process that is consistent with empirical estimates in the literature and captures the level of roughness of short-maturity option volatility. Even if the rough 1/2 component of the model has a low weight, our analysis of option data still identifies a level of roughness similar to that identified by the rough Heston model. Our large-scale empirical analysis on the S&P 500 and VIX markets shows the practical relevance of our model.