Is joint Kolmogorov Complexity order invariant?
You don't need symmetry of information. The invariance theorem does the trick. Let $p$ the smallest program such that $U(p) = \langle x, y\rangle$. One way of producing $(y, x)$ is to take make a program $q$ that runs whatever program it is given as input, interprets the output as a pair, and flips the two parts. This gives you a program $\overline{q}p$ to produce $\langle y, x\rangle$. Since $\overline{q}$ is only a constant number of bits long, the two $K$'s are equal up to a constant.

Now for higher numbers of arguments, the same idea works in principle. However, the constant does depend on the number of arguments: if I want to write a $q$ program to compute some $n$-tuple of numbers and re-arrange it into an arbitrary order, I need to encode the order in $\log(n!)$ bits (if the ordering is random). So if the number of arguments is not fixed to a constant, you need to take that into account.

Isn't there some other way, besides the $q$ program? No, if there were we could encode free information in the ordering of the tuple.

Assume for a contradiction that $K(x_1, \ldots, x_n)$ is invariant up to a constant to permutation of the arguments, with the constant independent of $n$. Let $X = \langle x_1, \ldots, x_n\rangle$ be the enumeration of the first $n$ binary strings, so that $K(X) \leq_+ K(n)$. Let $z$ be a random string with $|z| = \log(n!)$ Let $\langle x_1, \ldots, x_n \rangle$ be a random string. Index all permutations of $n$ elements by binary strings and pick the one corresponding to $z$. Call this permutation of our tuple $X_z$. Let $p$ and $p_z$ be the shortest programs for $X$ and $X_z$ respectively. Build a program $q$ that reads input $\overline{p}p_z$ and returns $z$. Putting all this together gives us 
$$\log(n!) \leq_+ K(z) \leq_+ |\overline{q}\overline{p}p_z| \leq_+ 2K(X) \leq_+ 2K(n) \leq_+ 4\log(n).$$