Incomplete sentences make it difficult, such as the OP.

Your example is great:

• Me jumped in the pool.
• I jumped in the pool.

The second sentence is correct, so you would go with, “My twin and I jumped in the pool.”

Similar example:

• The rain got me and my twin soaking wet.
• The rain got my twin and I soaking wet.

Simplify them:

• The rain got me soaking wet.
• The rain got I soaking wet.

The first sentence is correct, so go with, “The rain got me and my twin soaking wet.”