Vector spaces as bimodules The usual definition of a vector space $V$ over $K$ is as an abelian group, on which $(K\setminus\{0\},\cdot)$ acts on the left, such that furthermore the operation of $K$ on $V$ is compatible with the addition, i.e. we have $k\cdot (u + v) = k\cdot u + k\cdot v$ and also $(k + s)\cdot u = k\cdot u + s\cdot u$.
Now with the definition $u\cdot k := k\cdot u$ we have a right action (for which commutativity of $(K\setminus\{0\}, \cdot)$ is essential, as $u(ks) = (ks)u = (sk)u = s(ku) = s(uk) = (uk)s$), which makes $V$ into an $K-K$-bimodule as $(ku)s = s(ku) = (sk)u = (ks)u = k(su) = k(us)$.

But does there exists a vector space $V$, which has two different action of some field $K$ on $V$ which are not compatible, i.e. the order of application of two different elements matter, or equivalently does there exists a vector space, on which $K$ acts also on the right with similar laws, such that $V$ is not a $K$-$K$-bimodule w.r.t. to both actions?

 A: Check my profile for my dissertation, you'll find an example. Specifically, let $G$ be a group, $V$ a vector space on which $G$ acts on the left, and $F(V)$ the field of fractions of the symmetric algebra of $V$. Define a ring $R$ to be the left $F(V)$-vector space with basis $G$ with right action given by (for $g\in G$, $f\in F(V)$)
$$gf=(g\cdot f)g$$
Multiplication of elements of $G$ uses the product in $G$. $R$ is called a "skew group ring" of $G$ over $F(V)$ because it's the group ring twisted by an action on the coefficient ring.
Here the actions are compatible in the sense that $R$ is indeed a bimodule, but the left and right actions are different.
For an example, let $x_1,x_2,\ldots,x_n$ be a basis for a vector space, and let the symmetric group $S_n$ (the set of bijective functions from $[n]$ to itself) act by $w\cdot x_i=x_{w(i)}$. $S_n$ acts on the field of fractions in these indeterminates by, for example, 
$$w\cdot \frac1{x_i-x_j}=\frac1{x_{w(i)}-x_{w(j)}}$$
Then in our ring $R$ (which is not the module coefficient ring, which is $F(V)$, but rather the module) which contains $F(V)$ as a subring, has an action of $F(V)$ by right and left multiplication, where, for example,
$$wx_i=x_{w(i)}w$$
