Currently using Atahualpa theme for this site: hoxtonindustries.com.au
On my iPad and iPhone, the layout resizes nicely to fit either vertical or landscape screen - great! But the client would like a DIFFERENT LAYOUT for display on mobile phones. A good example of what we want to do is similar to this site: www.sunscreenpatios.com.au which places navigation blocks vertically and is much easier and more visual to navigate on a small screen.
Is it possible to do that with Atahualpa theme, or can anyone recommend a suitable plug-in?

there is a mobile template that is part of the jetpack plugin

Got it! Thanks. Now I just need to figure out the navigation menu - on iPhone, a menu in list form appears momentarily then disappears, making way for a mobile-specific display of the first page, when Menu+ button appears but it does not do anything??

you might want to ask about that on teh WordPress.org forum. I know nothing about how the jetpack plugin works internally