There are always two points on opposite sides of the Earth with the exact same temperature. And we can prove that.

Temperature changes continuously. If a and b are on opposite sides of the equator and D(a) = T(a) – T(b) is positive, then D(b) = T(b) – T(a) is negative. That means there must be some point x on the equator where D(x) = 0. At that point the two opposite sides are the same temperature.

Mathematicians call this the Intermediate Value Theorem which means if there is a continuous function that changes from of a positive value to a negative value (or the other way around) then it must, at some point, pass through zero.