Note that the property we're talking about isn't really irreversibility of individual transactions. We can come up with whatever policy about what can show up in a ledger, including "undo" transactions.

But what we want is that once transactions are recorded to a ledger, that it's not possible to remove that record.

This durability is a highly valuable property for financial transactions and other things where non-repudiation is desired. However, it's a possibly scary property, too (what if something copyrighted or illegal ends up in the forever-durable and public ledger?)