Solution

Mirror image is front-back inverted, but we often say that it is left-right inverted because we fix the up-down direction and use it together with the front-back direction to define the left-right direction. More precisely, if we consider $\hat\imath$ as up and $\hat\jmath$ as front, then left is defined as $\hat\imath\times\hat\jmath=\hat k$. When mirror image is front-back inverted, left becomes $\hat\imath\times(-\hat\jmath)=-\hat k$, and we say that the image is left-right inverted.

Imagine what happens when we fix the left-right direction and use it together with the front-back direction to define the up-down direction. If we consider $\hat k$ as left and $\hat\jmath$ as front, then up is defined as $\hat\jmath\times\hat k=\hat\imath$. When mirror image is front-back inverted, up becomes $-\hat\jmath\times\hat k=-\hat\imath$, and we might as well say that the image is up-down inverted.