  We explore the ground-state phase diagram of the $S\!=\!1/2$ two-leg ladder
with different isotropic leg interactions and uniform anisotropic rung ones,
which is described by the Hamiltonian ${\cal H}=J_{{\rm l},a}
\sum\nolimits_{j=1}^{L}{\vec S}_{j,a}\cdot {\vec S}_{j+1,a}+J_{{\rm l},b}
\sum\nolimits_{j=1}^{L} {\vec S}_{j,b}\cdot {\vec S}_{j+1,b}+J_{\rm r}
\sum\nolimits_{j=1}^{L} \bigl\{S_{j,a}^x S_{j,b}^x + S_{j,a}^y S_{j,b}^y +
\Delta S_{j,a}^z S_{j,b}^z \bigr\}$. This system has a frustration when
$J_{{\rm l},a} J_{{\rm l},b}\!<\!0$ irrespective of the sign of $J_{\rm r}$.
The phase diagrams on the $\Delta$ ($0\!\leq\!\Delta\!<\!1$) versus $J_{{\rm
l},b}$ plane in the cases of {$J_{{\rm l},a}\!=\!-0.2$ and $J_{{\rm
l},a}\!=\!0.2$ with $J_{{\rm r}}\!=\!-1$ are determined numerically. We employ
the physical consideration, the level spectroscopy analysis of the results
obtained by the exact diagonalization method and also the density-matrix
renormalization-group method. It is found that the non-collinear ferrimagnetic
(NCFR) state appears as the ground state in the frustrated region of the
parameters. Furthermore, the direct-product triplet-dimer (TD) state in which
all rungs form the TD pair is the exact ground state, when $J_{{\rm
l},a}\!+\!J_{{\rm l},b}\!=\!0$ and $0 \leq \Delta \leq 0.83$. The obtained
phase diagrams consist of the TD, $XY$ and Haldane phases as well as the NCFR
phase.
