The long-standing observations that different amorphous materials exhibit a pronounced enhancement of viscosity and eventually vitrify on compression or cooling continue to fascinate and challenge scientists(1), on the ground of their physical origin and practical implications. Glass formation is a generic phenomenon, observed in physically quite distinct systems that encompass hard and soft particles. It is believed that a common underlying scenario(2,3), namely cage formation, drives dynamical arrest, especially at high concentrations. Here, we identify a novel, asymmetric glassy state in soft colloidal mixtures, which is characterized by strongly anisotropically distorted cages, bearing similarities to those of hard-sphere glasses under shear. The anisotropy is induced by the presence of soft additives. This phenomenon seems to be generic to soft colloids and its origins lie in the penetrability of the constituent particles. The resulting phase diagram for mixtures of soft particles is clearly distinct from that of hard-sphere mixtures and brings forward a rich variety of vitrified states that delineate an ergodic lake in the parameter space spanned by the size ratio between the two components and by the concentration of the additives. Thus, a new route opens for the rational design of soft particles with desired tunable rheological properties.