  The key to unraveling intriguing phenomena observed in various Kitaev
materials lies in understanding the interplay of Kitaev ($K$) interaction and a
symmetric off-diagonal $\Gamma$ interaction. To provide insight into the
challenging problems, we study the quantum phase diagram of a bond-alternating
spin-$1/2$ $g_x$-$g_y$ $K$-$\Gamma$ chain by density-matrix renormalization
group method where $g_x$ and $g_y$ are the bond strengths of the odd and even
bonds, respectively. The phase diagram is dominated by even-Haldane ($g_x >
g_y$) and odd-Haldane ($g_x < g_y$) phases where the former is topologically
trivial while the latter is a symmetry-protected topological phase. Near the
antiferromagnetic Kitaev limit, there are two gapped $A_x$ and $A_y$ phases
characterized by distinct nonlocal string correlators. In contrast, the
isotropic ferromagnetic (FM) Kitaev point serves as a multicritical point where
two topological phase transitions meet. The remaining part of the phase diagram
contains three symmetry-breaking magnetic phases. One is a six-fold degenerate
FM$_{U_6}$ phase where all the spins are parallel to one of the $\pm \hat{x}$,
$\pm \hat{y}$, and $\pm \hat{z}$ axes in a six-site spin rotated basis, while
the other two have more complex spin structures with all the three spin
components being finite. Existence of a rank-2 spin-nematic ordering in the
latter is also discussed.
