We theoretically explore a scheme for generation of bright circularly and elliptically polarized high-order harmonics by bursts of linearly polarized pulses with a rotating polarization axis. Circularly polarized harmonics are formed if the bursts are comprised of N
pulses that uphold an N
-fold rotational symmetry, for N >
2. Rotating the polarization axes of the comprising pulses can generate elliptical harmonics with a collectively tunable ellipticity, from circular through elliptic to linear. The method preserves the single-cycle, single-atom and macroscopic physics of ‘standard’ linearly polarized high harmonic generation, with a high yield and cutoff energy. We investigate the method from a time-domain perspective, as well as a photonic perspective, and formulate the energy and spin-angular momentum conservation laws for this scheme. We find that the case of N
= 4 is optimal for this method, resulting with the highest conversion efficiency of elliptical photons. The new features of this source offer new applications to helical ultrafast spectroscopy and ellipsometry.