If $E\to M$ is a fiber bundle, it's convention to write 1. $E_p$ for the fiber at $p$, 2. $E^*$ for its dual bundle.
Exceptionally, for the tangent bundle $TM\to M$ almost every author writes 1. $T_pM$ for the fiber at $p$, 2. $T^*M$ for its dual bundle, instead of $TM_p$ and $TM^*$. Is there any reason for why this is the convention?