Let $n \geq 3$ be a positive integer representing the number of sides of a regular $n$-gon. Our objective is to find the maximum number of diagonals we can select such that any two selected diagonals either do not intersect within the interior of the $n$-gon or are perpendicular to each other.

Approach:

To explore this problem, we need to examine how diagonals interact with each other when drawn inside a regular polygon.

1. **Diagonals in an $n$-gon:**

A diagonal in a polygon is a line segment connecting two non-adjacent vertices. For a regular $n$-gon, each vertex connects to $n-3$ other vertices with diagonals, as it cannot connect to itself or its two adjacent vertices with a diagonal.

2. Number of Diagonals:

The total number of diagonals $D$ in a regular $n$-gon can be calculated using the formula:
$$D = \frac{n(n-3)}{2}$$

3. Conditions for Non-Intersecting or Perpendicular Diagonals:

- Two diagonals intersect in the interior if they form an "X" shape, meaning their lines extend and meet at a point inside the polygon.
- In a regular $n$-gon, diagonal pairs that do not intersect inside the interior (without being perpendicular) can be managed by strategic selection.
- Trying to construct (or analyze) the configurations shows that the number of non-intersecting diagonals without viewing their perpendicularity depends on the parity and structure of $n$.

4. Determining the Maximum Selection:

- **Odd $n$:** If $n \equiv 1 \pmod{2}$, then one extra condition arises that limits the number of selectable diagonals due to overlap considerations. Here, the maximum number of diagonals that satisfy the conditions is:
$$n - 3$$

- **Even $n$:** If $n$ is even, the structure allows for selecting an additional diagonal without causing intersection (since perpendicular conditions fit more neatly within the framework). Hence, for even $n$, the maximum is:
$$n - 2$$

Thus, the formula for the maximum number of diagonals that can be selected such that they do not intersect in the interior or are perpendicular is:
$$\boxed{n-3 \text{ if } n \equiv 1 \pmod{2}, \text{ and } n-2 \text{ otherwise}.}$$