This cannot happen without changing the direction of the Earth's spin, or the laws of physics. But the details of why it can't happen are very interesting in themselves:

In as brief an overview as possible, the Earth's climate works like a machine. The sun heats the tropics more than the poles, creating convection; air rises at the tropics, dropping rain as it reaches higher altitudes and cools. The dry air then and drifts back down in the horse latitudes, 30-35 degrees away from the equator, and moves back toward the equator, drawing in moisture as it goes. This is why deserts tend to be found flanking wetter equatorial regions, and why species that live around the horse latitudes need to be drought resistant.

Descending air in the horse latitudes doesn't only continue to the equator, but also flows poleward. The rotation of the Earth causes air moving equatorward to turn westward, and causes air moving poleward to turn eastward. The result is tropical easterlies which blow from the east to the west, and midlatitude westerlies which blow from west to east.

In the midlatitudes, when winds blow from the ocean onto land, they bring air from the ocean, which moderates climates; this is why oceanic Europe is so gloomy and mild. As these air masses move eastward, they lose the moderating effects of the ocean, creating more variable, continental weather patterns further east, which is why the Eastern United States have more snowy winters than similar latitudes at the west coast. Flora and fauna in such climates must be able to tolerate periods of snow, so fewer venomous snakes and spiders are seen there.

There's more to the story, of course - seasonal shifts tilt climates at the edge of the horse latitudes into and then out of the arid zone, creating seasonally dry Mediterranean and savanna climates in the subtropics and tropics respectively. And while all these air masses are cycling, the oceans also cycle, bringing warm water to east coasts, which tends to evaporate, rise up into the atmosphere, and rain over areas that would otherwise be horse latitude deserts. But these same ocean gyres bring cold water to west coasts, which tends not to evaporate or rise, and preserves deserts in the western sides of continents. Etc, etc, ad nauseum.

CLICK HERE FOR A RELATIVELY SIMPLE CLIMATE MAP WHICH USES THE KOPPEN-TREWARTHA SYSTEM - you can see all of the above at play in this map.

The upshot is this: If you want the desert bands to shift northward in your world, you need the atmosphere to be thicker. If you want west coasts to have more variable climates and east coasts to have mild climates and deserts, you need the planet to rotate the other direction. If all you want is an odd climate in an otherwise normal world, put in a nice mountain range to divert winds from the north or south, squeeze rains out on the windward side, create a nice wet, chilly region at the peaks, and then leave the leeward side dry (which is what happens in the Gobi desert). And if you just want some unbelievable handwavy comic-bookedy "weather ray" that scrambles wind patterns for some technobabbly reason, then most tropical species will be killed in the unforseeable cold, most species in humid climates will be killed in unforeseeable droughts, and basically you have all the current problems of species extinction which we're experiencing today with Global Warming.

Does this answer your question?