This question is related to How to make a sandwich from just one piece of bread?, asked on Feb 23 '17 by erz, and it goes as follows:

Question.If a convex closed and bounded region $C$ in the plane $\mathbb{R}^2$ can be cut along some straight line into two congruent pieces, must $C$ have either axial or central symmetry?

To be specific about what $cutting$ means, a $piece$ consists of the closure of the set of all points of $C$ that lie on the same side of the cutting line. Obviously, if $C$ has either axial or central symmetry, then it can be cut so.

The question can be phrased generally in $\mathbb{R}^n$, where we would cut a convex body along a hyperplane and consider $n$ kinds of symmetry, for example central, axial, and mirror in $\mathbb{R}^3$. I believe the answer is $yes$. Is it perhaps known already?

In retrospect (see Wlod AA's very nice answer): my intuition was way off.