Can I send the same old passport off twice to be renewed?

If you're talking about a British passport, no, I'm pretty sure you have to send your current or most recent one if you want to renew it.

And in response to the other answer, I've always received my old British passport back when renewing, but with the corner cut off to indicate it is expired. The process may be different for passports issued by other countries.

By the way, an old passport may still be used as photo ID (e.g. for domestic flights) even if it's no longer valid for international travel. Of course, if the photo is of you in the 1970s then it's no good, but if it's still fairly recent then it's probably fine.

No - the rules on the passport office website make it clear that if renewing a passport you must send one that has recently expired or is about to expire. There is a separate system for obtaining a new passport if you have lost yours. I am surprised they sent you the old one back - the website says they destroy old passports sent in with renewal applications.