Antipodes map in QGIS

What is relatively simple is to extract the vertices of any geometry and than change (switch) the coordinates to it's antipode position using this formula/expression:

make_point (
    if (
        x<0, 
        x($geometry)+180, 
        x($geometry)-180
    ),
    -y ($geometry)
)

See at the bottom of this answer for an explanation of the formula.

If you just want to create the antipode of a point you click on the map canvas, create a new point layer (be sure to have the CRS in EPSG:4326 for WGS84 projection), add a new Symbol Layer / Geometry generator / Geometry type: Points and paste the expression from above. Now every time you click on the map canvas to create a new point (blue in the following screenshot), a second point (the antipode, red) will automatically be added: point 2 (New Zealand) gets the antipode in Spain, 4 in Australia gets the antipode in the Atalantic, 5 in China in Argentinia and so on: every time you click, the antipode will be generated automatically:

To create a whole world antipodes map, here is a little bit more detailed description (see the steps on the next screenshot):

1. In QGIS, load a polygon of the world - I just introduced world to the coordinates field in the status bar to load the pre-installed World Polygon layer.
2. Go to Menu Vector / Geometry Tools / Extract vertices. You get a new layer with all the vertices. In case you got a multipoint layer, apply Multipart to singleparts (in the same menu as before).
3. Now, change the Symbol layer type of the extracted vertices from Simple Marker to Geometry generator / Geometry type: Points and add the expression from above.

What you get is the antipodes map you see on the next screenshot where I also added OpenStreetMap as additional basemap. The antipodes consist of vertices (red dots) - you could create lines and from these polygons again if you wish: Australia is on the Central Atlantic; Africa in the Central Pacific; India and Central Asia on the Southern Pacific (west of Southern America); North America on the Southern Indian Ocean; New Zealand over the Iberian Peninsula; South America stretching from Indonesia to China, Mongolia until eastern Siberia; and many small islands of the Pacific are dispersed all over Africa:

The formula is simple: the latitude just changes to its negative value (-y): 10 degrees North result in 10 degrees south. For longitude, it's x-value +/- 180 degrees, depending on it's original position East or West of the 0 degree meridian (London Greenwich). Thus: if x is smaller than 0 (= degrees West), add 180 degrees, if it is to the East of Greenwich (x has a positive value), substract from it 180 degrees. That's what the expression above does.