Aviation has a hard ceiling that no engine can raise: on a sphere, no two points can be further apart than half the circumference. For Earth that is about 20,015 km — the distance to your antipode. Every route ever flown is some fraction of that number.
The longest scheduled nonstop flights in recent years — Singapore to New York, around 15,300 km — cover roughly three-quarters of the antipodal maximum, staying airborne for 18-plus hours. Planned "Project Sunrise" routes such as Sydney–London would push past 17,000 km. But a true antipodal city pair like Auckland–Madrid (about 19,600 km apart) has never had a nonstop connection, and none is on the horizon.
Fuel is the tyranny. Every extra hour aloft means carrying fuel to carry fuel — the relationship is viciously non-linear, and past a certain range each additional kilometre costs disproportionate payload. Airlines only operate ultra-long-haul where a premium-heavy market pays for it. Antipodal pairs, almost by definition, are exotic: they link places whose historical and economic ties are thin precisely because they are as far apart as places can be.
Antipodal geometry has a quirk pilots would enjoy: between exact antipodes, every great circle is the same length. A flight from Madrid to its New Zealand antipode could depart heading north over the pole, south over Africa, east over Asia or west over the Atlantic — all identical distances on paper. In practice winds decide: eastbound routes ride the jet streams, which is why "round the world" record attempts fly west to east.
Fly to your antipode now and you will connect through a hub, typically covering 21,000–24,000 km of actual track over 26–35 hours. For the Spain–New Zealand corridor — the closest thing Europe has to an antipodal commute — a Madrid–Auckland itinerary via a Gulf or East Asian hub is about as antipodal as commercial travel gets. When Antigea shows you a result, the flight-time estimate uses greater-circle distance plus typical routing overhead; for a truly antipodal destination it honestly reports that no single flight covers it.
One consolation for the ultra-long-haul weary: at 20,015 km, you can stop checking whether there was a shorter way. There wasn't. Anywhere further would be on the way back.