Two way matching compares the invoice against the purchase order and leaves out the goods receipt. For purchases with nothing to deliver it is exactly right, because there is no receipt to be had. As a general default it removes the only routine evidence that anything actually arrived, and organisations usually drift into it because receipts were hard rather than because it fit.
Where the fit is genuine
Services, subscriptions, professional fees, rent, licences, maintenance contracts. Nothing arrives at a loading bay, so the only meaningful comparison is against what was agreed. What should sit alongside it is a confirmation from whoever commissioned the work that it happened, recorded distinguishably from the approval.
Where it is a silent compromise
Physical goods. The receipt is the only routine evidence of delivery, so without it an invoice for goods that never arrived looks exactly like one for goods that did. This is a real risk being accepted, and the problem is not accepting it but doing so without ever having decided to.
How to choose per purchase
Which match applies is a property of what was bought, not a global system setting. Configure it by category so goods get three-way and services get two-way, and review anything that has drifted from one to the other. A system that only supports one is forcing a choice that should not be uniform.
Questions people ask about two way matching
Is two way matching quicker?
Marginally, and speed is not the reason to choose it. The reason is that some purchases have no delivery to record.
What if we cannot collect receipts for goods?
Then say so explicitly and treat it as a known gap with compensating checks, rather than letting it become the unexamined default.
Can auditors object?
For goods they may reasonably ask how delivery is evidenced. Have an answer that is not simply that nobody records it.