row and column majored rotation matrix pre- or post- multiplied
Okay let us start by pointing out that a colmun major matrix is the same as a transposed row major matrix. So:

$$
{M_{cm}}^{T} = M_{rm} \tag{1}
$$

Then notice that matrixes have following properties. Matrix multiplication is associative (2a) and that the distribution of transpose reverses computation order (2b).

$$
A(BC)  =  (AB)C \tag{2a}
$$
$$
(AB)^T  =  B^TA^T \tag{2b}
$$

From these follow that the chain of multiplication events can be as long as one likes and all transpose does is inverse the computation. So given that we can conclude 

$$
{T_1}\ {T_2}\ \ldots\ {T_n}\ \vec{v} = \vec v\,^T\ {T_n}^T\ \ldots\ {T_2}^T\ {T_1}^T, \tag{3}
$$

so calculating in column major order is the same calculation as calculating in row major orientation. The difference is that the order of computation is reversed. An interpretation of this is that the difference is the same as looking at the object from the outside or looking at the situation from the object out.

Formula 3 should provide you the insight. Just one more thing a column vector is the same as a transpose of a row vector.

Note: You are allowed to mix and match the computations as much as you like. All you have do is remember is that the other approach is just transposed.