In a moment of carelessness, I mistakenly used one of my wallet's sending addresses as the recipient for a transfer from a bitcoin exchange. The exchange confirms the transaction completed, but nothing shows up on my end. Is there any way to recover the amount?
Bitcoin payments are irreversible
Any transaction issued with Bitcoin cannot be reversed, they can only be refunded by the person receiving the funds. That means you should take care to do business with people and organizations you know and trust, or who have an established reputation. For their part, businesses need to keep control of the payment requests they are displaying to their customers. Bitcoin can detect typos and usually won't let you send money to an invalid address by mistake. Additional services might exist in the future to provide more choice and protection for the consumer.
No, they might not be lost. Addresses can both be used both for sending and receiving transactions. You have sent them to one of your own addresses. In your case you transferred to another address you own. So your wallet will have a new total minus the transaction cost. So technically, you could refund yourself again but you will loose another transaction cost.
Crucial note: All of this depends on your wallet. Some online wallet services share addresses between different users, so you want to be careful if you are using one of those.
If you are using the standard client, all your addresses are fully under your control and are safe to receive transactions.