is it possible to send transactions back and forth between all 3 address types (legacy, segwit, native segwit-bech32)?
Or is one of them not able to send to another?
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Each address type is backwards compatable with the previous one. So a segwit address has all of the benefits of the previous versions built into it and knows how to utilize them. A legacy address doesn't know about any of the segwit code, and cannot benefit from any of the improvements.
This all get's tied together at the transaction level. Sending segwit -> segwit you get all of the benefits. Sending legacy -> segwit you don't get any, as the transaction can't support information coming from the legacy address to implement it into any segwit functions.
That being said, older clients may not recognize newer versions of addresses as valid, and won't let you send to them (as a precautionary measure to prevent users from sending to a non-bitcoin address) if you manually broadcast the transaction on the bitcoin network itself, it would send.
At a protocol level, they are all compatible. Transactions can spend any of them, and send to any of them.
Wallet software may of course have restrictions, but these are usually not about combinations. E.g. some wallets may be unable to generate a p2sh-segwit address to receive on, or be unable to send to bech32. However, I have not heard about software that puts restrictions on where it can send to based on what is being spent, for example.
There are no restrictions on sending from any type of outputs to any address type in the Bitcoin protocol, but some older wallets might not support sending to newer address types.
Let's take a better look at what happens for a Bitcoin transaction coming to pass:
In any case, issues with sending to any specific address type are caused by missing functionality in the sender's wallet and in no way related to the input types used.