> If you reflect every point in the plane across a given “axis” point, you get a half-turn rotation.
> Reflecting across a point is also the same as reflecting twice across two perpendicular lines which intersect at that point.
My claim was rather that "reflecting across a point" has no meaning by itself—in the strict, mathematical sense according to which a reflection has a fixed hyperplane https://en.wikipedia.org/wiki/Reflection_(mathematics) —unless you use one of these two statements as the definition. (Probably you'd better use the second, because the first doesn't generalise to higher dimensions.)
> Reflecting across a point is also the same as reflecting twice across two perpendicular lines which intersect at that point.
My claim was rather that "reflecting across a point" has no meaning by itself—in the strict, mathematical sense according to which a reflection has a fixed hyperplane https://en.wikipedia.org/wiki/Reflection_(mathematics) —unless you use one of these two statements as the definition. (Probably you'd better use the second, because the first doesn't generalise to higher dimensions.)