We consider a general form for the asymptotic Yang-Mills gauge potential in the four-dimensional Euclidean space. This form of the potential describes the vacuum configuration, and the Pontryagin index is the product of two arbitrary integers $m$ and $n$, $m$ denoting the covering through the four-dimensional polar angle and $n$ specifying the covering through the three-dimensional azimuthal angle. Thus the asymptotic form is suitable to discuss multi-instanton solutions. We prove that this asymptotic form is spherically symmetric by finding explicitly a smooth gauge transformation that compensates any given arbitrary rotation. In the context of a class of symmetries of physical interest, we discuss a systematic procedure of starting forms or Ans\"atze for the gauge potentials which are smooth everywhere and which have the general asymptotic form. We prove that the simplest Ansatz in which the asymptotic form is multiplied by an appropriate function of the four-dimensional distance gives a solution only when $m=n=1$. The solution in this case is the well-known Belavin-Polyakov-Schwarz-Tyupkin solution and it is spherically symmetric. In order to explore the symmetry properties of solutions other than $m=n=1$, we study Witten's multi-instanton solution. In particular, we examine his solution for two instantons when the position parameters of the two instantons coincide. The two scale parameters reduce to one in this case. Contrary to naive expectation, this solution which may be described as a doubly charged single instanton, is not spherically symmetric.