Please solve the programming task below using a self-contained code snippet in a markdown code block.

You are given an m x n binary matrix grid. An island is a group of 1's (representing land) connected 4-directionally (horizontal or vertical.) You may assume all four edges of the grid are surrounded by water.
An island is considered to be the same as another if they have the same shape, or have the same shape after rotation (90, 180, or 270 degrees only) or reflection (left/right direction or up/down direction).
Return the number of distinct islands.
 
Please complete the following python code precisely:
```python
class Solution:
    def numDistinctIslands2(self, grid: List[List[int]]) -> int:
```