Let's start with something a little simpler, like the constellations of the zodiac. These constellations are all near the celestial equator.
When viewed from moderate latitudes in the northern hemisphere, a person will typically face South, and look up a little bit to see whichever zodiac constellation is highest in the sky.
But when viewed in moderate latitudes in the southern hemisphere, a person will typically face North and look up a bit.
So yes, the constellation will be flipped upside down (due to the facing North vs. facing South).
But the constellation Crux is really far South. People in moderate latitudes in the northern hemisphere won't be able to see Crux ever. Similarly, people in moderate latitudes in the southern hemisphere won't be able to see the Big Dipper (part of Ursa Major). Ever.
Even people in moderate latitudes in the southern hemisphere may need to face at least a little South when viewing Crux. (People in the northern hemisphere can't see it at all -- it's below their southern horizon).
But if you're deep into the southern hemisphere, you can see Crux all year around, as it rotates around the southern celestial pole. This is similar to how people deep in the northern hemisphere can see the Big Dipper all year around, as it rotates around the northern celestial pole. Sometimes it will look upside down compared to other times, depending on how far it is rotated around the pole.