In practical applications, invalid transactions can not be stored on blockchain because there are validating nodes who check for the validity and genuineness of the transaction. But, there a possibility that an invalid transaction can be stored on the blockchain when 51% (or more) of validating nodes approve an invalid transaction.
The blocks of a blockchain can not be re-organized and the transactions can not be reverted. The main intention/feature of Blockchain is that the data in it can not be altered and hence re-organization of blocks are is not possible.
So to answer to your question in a practical scenario,
- No, invalid transaction be stored on a Blockchain.
- You can not re-organize or revert transactions on a Blockchain.