Let's talk about atmospheric circulation
Image courtesy Slideplayer.com, and the slideshow would be well worth your visit (especially slide #3). Click on the image to get a bigger view. Click on the link above to visit the slideshow.
As you can see, circulation pulls air away from the equator, both to the north and to the south. Ignoring the dynamics of weather for a moment: you tend to get high pressures at the poles and near the equators and low pressure zones near the tropics (of Cancer and Capricorn). They move around for lots of reasons, but the point I'm making is that it's unlikely to get a high above the equator and a low below the equator at the same time at the same longitude because that's not what the air wants to do.
Curiously, we can replace your test case continent with open ocean, like the Pacific. It's a large, uniform "surface" that crosses the equator. If you can get the climatic condition you want with one, you can get it with the other. Bit of rambling there, back to the story....
Now, I say unlikely because the pretty picture isn't taking into account axial tilt, which shifts the "center" of where sunlight strikes on the planet. Depending on the season, the "center" is either slightly above or slightly below the planetary equator. That shifts things up and down (simplistically, very simplistically). I'm also not taking into account oceanic currents, which contribute to atmospheric flow nor the tendency of a planet to form a jet stream, which is rotational wind that slides around things because nothing is actually smooth. When you combine all these (and without doubt others) you have the ability to position a high on one side of the equator and a low on the other — but atmospheres are dynamic, so it wouldn't last because (and this is my opinion), it's not a stable condition. As mentioned above, you would expect a high/high above and below the equator and lows at the tropics.
Briefly (aka, days) you could have the situation you're looking for, but not for a long period of time.