When a component-wise operation is applied to two series, it is applied in the same way that, for example, two mathematical functions are added to form another function.

That is to say, the _labels_ are used to determine which values to add (or multiply, etc.) together. That means the the orderings of the labels in the two input series are not required to be the same. Each value is strongly tied to its label, and this connection will be maintained through the calculations.

The labels in the index of the resultant series will consist of the _union_ of the values in each of the argument series. If the index for one of the argument series contains the label X, but the other does not, then the result index will contain the label X, but its value will be set to NaN.

Regarding the ordering of the labels in the result, I have seen the following, commonsensical behavior. If $s1$ and $s2$ have the same index, then $s1 + s2$ will have that index. But if there is any difference whatsoever, then the result index will contain the union of the labels, and will be sorted by the canonical ordering for the type of the labels.

That is to say, the _labels_ are used to determine which values to add (or multiply, etc.) together. That means the the orderings of the labels in the two input series are not required to be the same. Each value is strongly tied to its label, and this connection will be maintained through the calculations.

The labels in the index of the resultant series will consist of the _union_ of the values in each of the argument series. If the index for one of the argument series contains the label X, but the other does not, then the result index will contain the label X, but its value will be set to NaN.

Regarding the ordering of the labels in the result, I have seen the following, commonsensical behavior. If $s1$ and $s2$ have the same index, then $s1 + s2$ will have that index. But if there is any difference whatsoever, then the result index will contain the union of the labels, and will be sorted by the canonical ordering for the type of the labels.