Intuition behind the asymmetry between invertibility of key and ciphertext in block ciphers I have studied the design of block ciphers like DES and AES. 

I know the actual algorithms, but I don't intuitively "grok" the most important property of them:

The most important property, is that if you have the key and the ciphertext, you can calculate easily the plaintext, but if you have the plaintext and ciphertext, you cannot easily calculate the key.

That is, the block cipher is a function $$f(P,K)= C$$ and the essential property is that the inverse w.r.t. P is very easily caclulated, but the inverse w.r.t. K is not.

I don't intuitively understand why this assymmetry is true, even though I know how to actually do the decryption algorithm. Is there an intuitive explanation of this asymmetry?
If you had a block cipher with this property, it would necessarily be insecure!

The most basic requirement for a block cipher is to be a pseudorandom function (PRF). In the game that defines security of PRFs, the attacker gets to choose input blocks $x_1, x_2, \ldots$ and gets back responses $y_1, y_2, \ldots$. It should be hard to distinguish whether the $y_i$'s are uniformly random or have the form $y_i = F(k,x_i)$ for a random (secret) key $k$.

Suppose your block cipher was ``key-invertible'' in the way you describe. Here's an attack against the PRF security of the block cipher. Ask for some $x_1$ and get back $y_1$. Now you do this key inversion process and you can learn (a candidate) key $k'$. Then you can make one more query on $x_2$ and get back $y_2$. If $y_2 = F(k',x_2)$ then you guess that you are getting outputs of the PRF; otherwise you guess that you are getting random responses. 

This attack completely breaks PRF security. It assumes that the key-inversion process is perfectly correct --- that is, given $x$ and $y = F(k,x)$ for unknown $k$, you can always compute that exact $k$. Even if you assume the key-inversion procedure is not always correct (e.g., maybe there are two keys that satisfy $F(k,x) = F(k',x)$ so sometimes you get it wrong), It seems likely to me that this attack can be extended.