Chiral molecules scatter circularly-polarized light at different rates according to their handedness and that of the incident photons. A right-handed molecule, for example, scatters right-handed circularly-polarized light at a different rate to lefthanded light. If the energy of the scattered photon is equal to that of the input, the chiroptical effect is known as Rayleigh optical activity, whilst if energy is imparted on to the material in the scattering process it is termed Raman optical activity (ROA). The vast majority of biomolecules are chiral, and ROA is a particularly important form of scattering as it underpins vibrational chiroptical spectroscopic techniques that are pivotal in determining their structures, conformations, and functionalities. Twisted beams of light that convey an optical orbital angular momentum (OAM) of ℓℏ per photon are also chiral, being able to twist either clockwise or anticlockwise along the direction of propagation, and this handedness is completely distinct from circular-polarization handedness. Here it is shown that a twisted beam of Laguerre-Gaussian light produces forms of Rayleigh and Raman optical activity that are sensitive to the direction that the beam twists, producing chiroptical effects dependent upon both the sign and magnitude of ℓ in both anisotropic and isotropic molecular systems. This circular-vortex differential scattering effect is seen to stem from electric-quadrupole transitions coupling to the gradient of the field. The scattered differential intensity is further developed to account for its distinct scattering angle and off-axis beam alignment dependencies, and prospective experimental scattering geometries are highlighted in which there is significant scope for enhanced optical activity signals using the OAM of twisted light.