Two way matching compares the invoice against the purchase order and stops there, with no goods receipt in the comparison. That is the correct control for some purchases and a real reduction in protection for others, and organisations frequently drift into using it universally because receipts are hard to collect rather than because it is appropriate.
When it is the right control
Services, subscriptions, professional fees, rent and anything else with no physical delivery to record. There is no goods receipt to be had, so the comparison is necessarily between the invoice and what was agreed, plus a separate confirmation from whoever commissioned the work that it was actually delivered.
When it is a compromise
Physical goods. Without a receipt, nothing in the routine process evidences that the delivery happened, so an invoice for goods that never arrived looks identical to one for goods that did. Using two-way matching for goods is a decision to accept that risk, and it is worth making it consciously rather than by default because receipting proved difficult.
The confirmation that should accompany it
For services, somebody who can say the work was done should confirm it, and that confirmation is the functional equivalent of a receipt. Where that person is also the approver, the two acts should be distinguishable on the record, since approving the spend and confirming delivery are different assertions.
Questions people ask about 2 way matching in accounts payable
Is two way matching acceptable to auditors?
For services it is standard practice. For goods, expect questions about how delivery is evidenced, and have an answer ready.
Can we use different matching by category?
Yes, and you should. Which match applies is a property of what was bought rather than a global setting.
What if receipts are simply never recorded?
Then you are running two-way matching for goods with the delivery risk unmitigated. Say so explicitly, so it is a decision rather than a drift.