Thank you for the kind words.

As to the software, you should realize that there are several kinds of symmetry. Even in the plane, one can reflect a shape in a point or in a line. More generally, one defines a group of transformations that leave an object (or a combination of objects) invariant. A 3D cube can be reflected in each of its elements: a node, an edge, and a face. It may be rotated around a node or an edge. In 4D, a tesseract can be rotated also around its 2D surface. When, the center/surface/body of reflection coincide with one of the coordinate constructs (e.g., x-axis, xy-plane, etc.) or is parallel to one, the formulas are pretty simple. You will realize this as soon as you start with a linear or vector algebra text. For a general configuration, the formulas require some fluency with multidimensional vector algebra. Any software that might exist will probably require of your to define that object in which reflection is sought.

I would check on Mathematica from Wolfram Research or MatLab.

All the best,
Alexander Bogomolny