1. **Problem Statement:** Describe the reflections and/or rotations that carry a square onto itself. 2. **Understanding the square's symmetry:** A square has several symmetries including rotations and reflections that map it onto itself. 3. **Rotations:** A square can be rotated about its center by multiples of 90 degrees and still coincide with its original position. These rotations are: $$90^\circ, 180^\circ, 270^\circ, 360^\circ$$ 4. **Reflections:** A square can be reflected about lines of symmetry that pass through its center. These lines are: - The vertical line through the midpoints of the left and right sides. - The horizontal line through the midpoints of the top and bottom sides. - The two diagonals of the square. 5. **Summary:** The square is invariant under 4 rotations (including the identity rotation) and 4 reflections (2 through midpoints of opposite sides and 2 through diagonals). 6. **Final answer:** The figure maps onto itself under rotations of $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$, and reflections about the vertical axis, horizontal axis, and both diagonals.