The linear response of turbulence to a distortion and simple rotation is investigated in this paper from a fundamental theoretical standpoint. Quadratic flows are a special case of planar flows with constant mean velocity gradients, which can be characterized by a constant rate of strain D and a constant spanwise (normal to the plane) vorticity component −W=2Ω0, with arbitrary values (here, we can take D>0 and W>0 without any loss of generality). According to the sign of D2−Ω02, streamlines are hyperbolic, rectilinear (pure shear flow) or elliptic. Since these flows can also be considered as mean flows, when superimposing a three-dimensional disturbance (or fluctuating turbulent) field which satisfies statistical homogeneity, the linearized analysis of the disturbance field is of interest both from the point of view of hydrodynamic stability (e.g., the elliptical flow instability) and from the point of view of homogeneous rapid distortion theory (RDT) including applications to the basic statistics. The case of quadratic flow in a rotating frame (with angular velocity Ω in the direction normal to the plane of the basic flow) is revisited in this paper, in order to complete—with the three parameters D, −W=2Ω0 and Ω—previous works on linear theory by Cambon et al. [J. Fluid Mech. 278, 175 (1994)] and Speziale, Abid, and Blaisdell [Phys. Fluids 8, 781 (1996)]. From a simplified “pressure-less” linear approach, a general stability criterion is derived based on the value of the modified Bradshaw number Br=[D2−(2Ω+Ω0)2]/S2 (S=D−Ω0), which coincides with the rotational Richardson number introduced by Bradshaw [J. Fluid Mech. 36, 177 (1969)] and denoted by B=2Ω(S−2Ω)/S2 (in particular, for the case of pure shear flow where D=W/2=−Ω0=S/2). It is shown that this criterion gives results identical to the “true” linear stability analysis (including the effect of the fluctuating pressure) if the absolute vorticity 2Ω+2Ω0 has a zero value. In addition, the relevance of this criterion is checked with respect to the true linear approach in distorted wave space and related RDT applications. For all of the cases, the maximum amplification for the three-dimensional disturbance field is found for zero tilting vorticity 2Ω+Ω0 and for pure spanwise modes (with wave vector normal to the plane of the quadratic flow), in accordance with the generalized Bradshaw criterion and other results in hydrodynamic stability. For other spectral directions, the agreement is not as complete except for the pure shear case, and this is particularly discussed looking at statistical RDT solutions, which involve a summation over all directions of the wave vector. Finally, the impact of the whole analysis on second-order, one-point modeling is discussed.