Is there an orthogonal time dimension in Minkowski spacetime? Space dimensions are orthogonal one to each other. But what about time in the Minkowski diagram?
At first sight, time seems to be orthogonal to space. But we have to consider that each Minkowski diagram is an observer's diagram. Thus, an object that is not moving with regard to the observer will describe a vertical worldline in the Minkowski diagram of the observer. However, from the point of view of another observer (inertial frame) the worldline described by the object might be inclined. Even the worldline of the observer himself which is the y-axis might be sloped from the point of view of other observers. 
The conclusion seems to be that time is not orthogonal to space. Or is there an error?
 A: Time is orthogonal to space. Check that $(1, 0, 0, 0) \cdot (0, 1, 0, 0) = 0$ and likewise for all other spacelike unit vectors, where $\cdot$ represents the Lorentz invariant scalar product and I put time in the 0th coordinate.
The inclination originates in the Lorentz transformation that you use to go from one observer's point of view to another one's. Perfoming a boost creates an $x$ dependence in $t'$ and a $t$ dependence in $x'$:
$$ t' = \gamma \left(t - \frac{v}{c^2} x\right), x' = \gamma (x - v t)$$
This also happens in regular euclidean space, e.g. when perfoming a rotation about the $z$ axis, giving
$$ x' = \cos\theta \ x - \sin\theta\  y,\quad y' = \sin\theta\ x + \cos\theta\ y$$
So after a rotation a previousely vertical line will look inclined as well.
Edit: Observe that there is no t′ dependence in x′, even after the transformation. So, while time is always orthogonal to space, an object at rest for observer $O$ (say the observer's nose) will not be at rest for an observer $O'$ that moves away from observer $O$ at a speed $v \neq 0$.
A: Any timelike vector has a 3-dimensional subspace orthogonal to it, orthogonal under the Minkowski inner product.  Such a subspace is spanned by vectors that are spacelike under the inner product, and all linear combinations of those vectors are spacelike.
Any timelike vector might be the direction a massive object follows at some point on its worldline.  The 3-dimensional subspace orthogonal to that four-velocity is what an observer following that trajectory would instantaneously consider "space".
An observer's "time" is always orthogonal to his "space", and further, all observers would agree that that is so.
