Why can I see sometimes a horizontal half moon instead of a vertical one?
It occurs because the lit half of the moon points towards the sun, along a great circle in the sky.  The half-moon will be 90 degrees away from the sun in the sky. So by (say) 8pm (at equinox for simplicity) the half-moon will be visible in the South west, but the sun will be below the horizon.  The great circle from the moon to the sun will go &quot;down&quot; relative to the horizon.
The angle that it points down will depend on where you are on Earth. Only near the equator could the moon be completely horizontal,