In short the answer is Yes and No, and here is why.
We all see the same moon phases more or less on the same day. In the Northern Hemisphere the waxing is from right to left and in the Southern Hemisphere it is from left to right. Each hemisphere sees a different orientation of the moon with respect to the horizon and to a small degree your time zone.
Our moon phases are caused from the reflection of the sun on the moons surface. A line called a terminal line by astronomers is created on the moons surface distinguishing the light and dark. It creates the waning, waxing, new and full moon phases that are seen by all on earth at the same time. This is dictated by the positioning of the moon to the sun and its light that reflects off the moons surface.
Below is an example of Northern and Southern Hemisphere moon phases.
Click on the link below to see an animation and a detailed explanation of the moon in different hemispheres.