Antipodal Coordinates

Two points are called antipodal if they are at two opposite points on a circle or a sphere. They are also said to be diametrically opposed or that they are the most distant points of each other on a spheroid.

Here is an image presenting the earth globe with the continents presented once in yellow, and a second time in blue but with their antipodal coordinates (source wikipedia).

The portions where the yellow and blue continents overlap, are locations on Earth, where digging a rectilinear hole, perpendicular to the ground and passing through the center of the earth would land on another continent.

— From the coordinates on a sphere, take the opposite value of the latitude (which is to put or remove a minus sign -), and remove or add 180 to the longitude (to keep it in the interval [-180; +180]) for GPS coordinates.

The antipodal coordinates $ (\theta', \phi') $ of a point $ (\theta, \phi) $ on a sphere are given by $ \theta' = \theta \pm 180^\circ $ and $ \phi' = -\phi $.

— From 3D Cartesian coordinates, take the opposite of each component (x, y, z) ie (-x, -y, -z)

— From an address/city, use a geocoding tool (allowing you to find the GPS coordinates of an address).

Lines of longitude (vertical) connect the antipodal points, following a line of longitude guaranteed to pass through the antipodal point around the sphere.

For its position, measure the distance to one pole from the original point, and transfer it to the other pole to find the antipodal point.

Very few cities are polar opposites, but there are La Coruña (Spain) and Christchurch (New Zealand) or Beijing (China) and Bahía Blanca (Argentina) or Taipei (Taiwan) and Asunción (Paraguay)