It's called 'Identity' in Thunderbird. See http://kb.mozillazine.org/Mozilla_Suite_:_FAQs_:_Mail_Aliases

Many thanks for your prompt reply. It gets me most of the way there.

Messages sent from my main hotmail address in TB to the Outlook alias appear in the sent folder but do not appear in any TB Inbox folder. They do appear in my Outlook Inbox. So is it possible to get TB to receive them into an Inbox?

Messages sent from my Outlook alias address in Outlook to my main hotmail address get through to TB's hotmail Inbox and also appear in my Outlook Inbox. This works as expected.

Messages sent from my main hotmail address in TB to the Outlook alias

So are you saying you're sending a message to yourself, not to the main account, but to an alias email address?

What is the account type for the main account - POP or IMAP?

What is the account type in Outlook?

So are you saying you're sending a message to yourself, not to the main account, but to an alias email address? Yes.

What is the account type for the main account - POP or IMAP? All IMAP and using the same server.

What is the account type in Outlook? https://outlook.live.com/mail/

Thanks again.

Do you receive any messages in Thunderbird for the Outlook alias at all?

Do you receive any messages in Thunderbird for the Outlook alias at all? No.

Are you certain your main Hotmail email address and your Outlook alias belong to the same Microsoft account?

Ahh, that's got it. I thought that the mail server would be the same for all Microsoft accounts, I should've known better :-).

Thank you for your support, much appreciated.