What you are describing is called line symmetry. A more precise definition is that if you draw a line of symmetry, each point on one side of the line has a corresponding point on the opposite side of the line. If you connect these two points, that line is perpendicular to the line of symmetry. If this tessellation did have a line of symmetry, we would refer to it as a reflection. A reflection is flipping a figure across a line and making a mirror image.
All reflections have mirror lines.
There are three other types of transformations in a plane. They are all transformations that move a pattern in a way that the pattern still matches the original pattern exactly. Let's look for the other three types. We could slide the whole pattern sideways. This kind of symmetry is called a translation.