Why the reals with the operation $x \bullet y = \sqrt[3]{x^3 + y^3}$ is a group? The operation above is a group for the real numbers, since 0 is the identity element, and the negative of any real number is its inverse, as it can be observed trivially. Associativity is less trivial, but it holds.
In fact, if we substitute 3 for any odd number (5, 7 ...), the operation satisfies the properties of the group. However, any even number fails.
Is there any geometric / analytic / ... interpretation why an operation like $x \bullet y = \sqrt[3]{x^3 + y^3}$ is associative and, as a consequence, it gives the structure of a group to the reals?

For an arbitrary bijection $f\colon \mathbf R \to \mathbf R$, the operation $x*y = f^{-1}(f(x) + f(y))$ is a group law on $\mathbf R$. All this says is that if you rename each real number $x$ as $f(x)$ then you can convert the original group law $+$ into a group law $*$ so that $f$ is an isomorphism from $(\mathbf R, *)$ to $(\mathbf R,+)$. The intuition is algebraic, not geometric. There is nothing magical about $n$th roots for odd $n$ other than being a bijection.
The hyperbolic tangent function $\tanh \colon \mathbf R \to (-1,1)$ is a bijection that lets you transport addition on $\mathbf R$ to a group law on $(-1,1)$ that is used in special relativity (addition of velocities in one-dimensional motion). The inverse of this bijection, up to a scaling factor, is called “rapidity” in physics.
