Tensor Index Ordering Whilst I agree that the order of tensor indices is important, $T_{ij} \ne T_{ji} $,
I'm wondering if changing the order of the covariant and contravariant indices relative
to eachother has any effect on the tensors, in other words; are the following tensors
equivalent?
$$
T_{i\ \  j\ \  k}^{\  a  \  b} \stackrel{?}{=} T_{ijk}^{\ \ \   ab}
$$
I'm mainly asking this from a practical standpoint, I realise that the first is
the component of a member of $V^*\otimes V\otimes V^* \otimes V \otimes V^*$, and that the second is a component of $V^*\otimes V^* \otimes V^* \otimes V \otimes V$, but since the
covariant and contravariant indices always seem to pair up with eachother
(when contracted, etc...) will any of the computations involving these
tensors be affected by assuming that all dual basis components come first, and original basis components come second? 
Most authors don't seem to pay too much attention to this, so is there anything wrong with always assuming this?

If you have a metric, and you allow yourself to use the shortcut of "raising an lowering indices", for the so-called musical isomorphisms then the order does matter
E.g., if you do raise and lower indices, you have
$$T_{i}{}^{a}{}_{j}{}^{b}{}_{k} = T_{imj}{}^{bd}g^{am}g_{kd} \ne T_{ijk}{}^{ab}$$
If you do not use that notation at all, then you simply define the positions of the indices the first time you introduce the tensor, and you need to be consistent with your choice.
