  We show that Floquet engineering using bicircular light (BCL) is a versatile
way to control magnetic symmetries and topology in materials. The electric
field of BCL, which is a superposition of two circularly polarized light waves
with frequencies that are integer multiples of each other, traces out a rose
pattern in the polarization plane that can be chosen to break selective
symmetries, including spatial inversion. Using a realistic low-energy model, we
theoretically demonstrate that the three-dimensional Dirac semimetal
Cd$_3$As$_2$ is a promising platform for BCL Floquet engineering. Without
strain, BCL irradiation induces a transition to a non-centrosymmetric magnetic
Weyl semimetal phase with tunable energy separation between the Weyl nodes. In
the presence of strain, we predict the emergence of a magnetic topological
crystalline insulator with exotic unpinned surface Dirac states that are
protected by a combination of twofold rotation and time-reversal $(2')$ and can
be controlled by light.
