Well, the clock is UTC. That's the whole concept :) The clock simply combines our local times with UTC in one clock face. To make UTC work for all time zones, it also rotates UTC according to your location on Earth. This means that the reading of time is the same everywhere at any time using the rotated UTC. The clock's algorithm also automatically takes care of all time changes, keeping the reading of time consistent globally.
We actually can't just use UTC as is and eliminate time zones completely. If we do so, people in some places would go to work at 2am, in other places at 8pm. That's even more confusing. So the clock of hTime aims to use the best of both local and global times worlds in one clock interface.
As for day-changing, I hear you. Again this is UTC, so what the day of UTC is is the day of hTime. Something to work more on indeed.