Direct order invoices in Xero are marked as paid once they have a payment against them; it's not just a status -- it fully depends on a payment existing.
When you reconcile an invoice and match it against the actual deposit in the bank feed, that will mark it as paid/authorized.
However, if your accountant or bookkeeper needs to have them sent as 'paid' without prior reconciliation with your bank, you may force Reconcile.ly to do so.
Although we honestly don't recommend having this option enabled, we trust your accountant knows how to make good use of the option.
NOTE: Please beware that enabling this option will completely skip reconciliation.
If more questions come up, let us know via live chat, and we will be glad to help!