Planet with ecliptic inclination of 180 degrees in the Solar System?

A 180° inclination would mean that it's on the exact same plane as the regular planets in the solar system, but in retrograde motion. I suppose anything is possible, but it's not likely that it got in that motion naturally. It might have entered the retrograde motion after an encounter with another massive object, which altered its orbital motion.

Now, we've been discovering more and more dwarf planets and comets in quite eccentric orbits around the Sun. For example, the dwarf planet Eris, which is currently the largest dwarf planet in the solar system (bigger even than Pluto). It has an inclination of 44° to the ecliptic, and it took us until 2005 to discover it. So weirdly inclined orbits are absolutely possible. http://is.gd/KGO4NT

There are some asteroids with about 180 degrees tilt of their orbit with respect to Earth's ecliptic and the ecliptics of other planets. It is extremely unlikely we would not have noticed as big as dwarf planet size, unless more than 100 AU from the Sun.

A captured rogue yes, it is possible. But anything Earth sized would have been detected by now. The interaction of bodies and orbits is well understood.

Anything is probable if not possible. In this case it will move in the usual opposite direction of other planets.

That would mean that they move retrograde around the sun. A few dozen asteroids are know like that. Here is the list (see link).