For $d$ odd these two components are disconnected, because $I$ and $-I$ induce opposite orientation. For $d$ even, you have an involution, which takes a lattice to a complex conjugate lattice. This involution takes a lattice $\tau_1, \dotsc, \tau_{2d}$ to a complex conjugate lattice $\bar\tau_1, \dotsc, \bar\tau_{2d}$, and has many fixed points.
$\DeclareMathOperator\GL{GL}$However, be very careful about this: "By this https://mathoverflow.net/q/126329 the moduli space of complex $d$-tori is $X = \GL_{d}(\mathbb{C})\backslash{\GL_{2d}(\mathbb{R})}/{\GL_{2d}(\mathbb{Z})}$": for $d\geq 2$, this double quotient is very non-Hausdorff, because $\mathbb{R})/{\GL_{2d}(\mathbb{Z})}$ acts on $\GL_{d}(\mathbb{C})\backslash{\GL_{2d}}$ with dense orbits, as follows from Ratner theory. It's not very good idea to consider this quotient as an orbifold, because it has codiscrete topology: all open sets are either empty or everything. Detailed explanation is found in Ergodic complex structures on hyperkahler manifolds.
