I had the exact same problem with my PS3, worked over scart but nothing on hdmi. 1st plug in the scart and the hdmi, then go into your Harmon's settings and there should be a menu/option that allows you to change the output from scart to hdmi. My ps3 was displaying dvd's at 720P until I turned on the ps3 upscaler. It would be nice if they auto configured, instead of looking through all the settings.
It's very very unlikely to be the tv. The tv auto switches inputs when you turn on another device, and then when you turn it off goes back to either the scart or digi tuner depending on how you set up the options on the tv. I have mine set to scart so it starts up with sky and not the digi tuner.
You can only get SD from a scart lead. You have to use hdmi for HD. The tv will of course scale an sd image to match the screens resolution automatically. But it's not the same as HD over HDMI.
Also you may find that the tv does a better job of upscaling dvd's than your player. Try leaving the scaler on your player turned off and see what it's like. Then if you have the option to use the players upscaler try that and see if the picture is better or worse. If that makes any sense. Basically the tv's upscaler may be better than your dvd players.