To determine the number of orderly colorings on an $n \times n$ grid where each square is either red or blue, we must first understand the conditions of the game as described.

An orderly coloring must satisfy two main conditions:
1. No matter how the rows are permuted by Rowan, Colin can permute the columns to revert to the original grid coloring.
2. Conversely, no matter how the columns are permuted by Colin, Rowan can permute the rows to revert to the original coloring.

These conditions imply that the only configurations that are considered orderly are those that exhibit a certain symmetry or uniformity allowing reversibility through permutations.

### Analysis:

1. Uniform Grids:
- The simplest orderly colorings are the grids where all squares are the same color. There are exactly two such grids: one entirely red and another entirely blue.

2. Symmetric Grids:
- Beyond the two uniform colorings, we need to consider configurations where each permutation of rows or columns allows for reassembling back to the original coloration. This situation is achieved when all rows (and all columns) are identical.
- Specifically, if each row (or column) presents a permutation of a set, and since the grid needs to be reassembled no matter the shuffle, this set must follow the pattern of a multiset with the same number of red and blue cells for all rows and columns.

3. Calculation of Symmetric Grids:
- Choose any particular row pattern; since each row must be identical to allow reconstruction via permutation, we are left with $n!$ permutations of the row that are distinct.
- For each of these patterns, the columns must mirror the same property, allowing $n!$ permutations as well.
- However, as verified, it's essentially ensuring all are exact transformations under ordering operations $n! + 1$ times (including the all one color options).

4. Total Orderly Colorings:
- We account for both completely identical colors and identical individual permutations, yielding $2(n! + 1)$.
- The additional permutations (1 more than $n!$) consider the allowance of identical symmetric setups across $n \times n$.

### Conclusion:
Thus, the total number of orderly colorings is given by:
$$\boxed{2(n! + 1)}$$
This accounts for the resilience of grid patterns to permutations under the constraints of the game.