Can I set my HTC Desire into sync-mode without using touch screen The screen of my HTC Desire just broke (It's not shattered, but it's completely black).
I have no reason to believe anything else is broken (When I start it up, I hear the short tune that plays when the logo is normally displayed). Now, I'd really like not to lose any data, so I'd like to sync it to a PC using USB; however, I've set the default mode to tethering.

Is there any way for me to set the phone to sync-mode or any other way for me to recover my data?

I know that I can extract the memory card and use a card reader for that data. This question is specifically about the data that doesn't reside on the extractable memory card.
If the device itself works as you suggest you might be able to access the recovery mode, enable USB and then download the files you need. However this will not be a USB sync access, instead it can be a fastboot or ADB access.

To do so take your Desire which have to powered off and press and hold Volume Down. Now start it up by pressing Power.

In the recovery mode you are able to enable USB access via ADB. As the recovery mode uses only Volume Down, Volume Up and Power it can be controlled even without seeing anything. 

You only have to know which keys have to be pressed in which order. This is of course the complicated part. I don't have an unrooted device with the original recovery here for giving you the orders which keys to press. May be you find an Youtube video showing it or a very detailed manual.