  We define, in $C_p$-equivariant homotopy theory for $p>2$, a notion of
$\mu_p$-orientation analogous to a $C_2$-equivariant Real orientation. The
definition hinges on a $C_p$-space $\mathbb{CP}^{\infty}_{\mu_p}$, which we
prove to be homologically even in a sense generalizing recent $C_2$-equivariant
work on conjugation spaces.
  We prove that the height $p-1$ Morava $E$-theory is $\mu_p$-oriented and that
$\mathrm{tmf}(2)$ is $\mu_3$-oriented. We explain how a single equivariant map
$v_1^{\mu_p}:S^{2\rho} \to \Sigma^{\infty} \mathbb{CP}^{\infty}_{\mu_p}$
completely generates the homotopy of $E_{p-1}$ and $\mathrm{tmf}(2)$,
expressing a height-shifting phenomenon pervasive in equivariant chromatic
homotopy theory.
