I've setup a number of searches on the website which use start and due date filters for TODAY and TOMORROW. These work exactly as I'd expect on the website and correctly returning the tasks I'm expecting to see based on the criteria each day. However, on my iPad and iPhone, these same searches show nothing.
I suspect that the iOS app is changing the relative date values and is setting it to an absolute - probably the date I created the filter and it was synced. More testing is needed to prove this.