Is it possible, if unlikely, that a planet of similar size to the Earth on a diametrically opposite orbit taking the same time could be hidden behind the sun without our knowledge?
The short answer is no. Earth's orbit is not circular, that is to say, if this planet was 180 degrees ahead or behind of us, it would eventually stop being hidden by the Sun, due to Kepler's second law. So if this planet was to stay hidden at all times, then its orbit would need to be symmetrical to Earth's orbit and not be sharing the same second focus or orbit path. Such planet could theoretically exist, but we would have knowledge of its existence with today's technology.
How would we detect this planet? Well, there are many possible ways.
This planet would interact gravitationally with the rest of the solar system and we would be able to detect these perturbations. In fact, we would be able to deduce this planet's existence just by looking at its effects on other planet orbits.
Finally, we actually have a satellite (one of the STEREO satellites) that "looked" behind the Sun. At the same time, we have many satellites and probes orbiting other planets around the Solar System. If such planet existed, then we would have seen it.