Is it possible to implement a data structure that has properties of both sortedness and orderliness?

Yes, such a structure exists. For example, in programming languages, there are data structures that preserve the order of insertion and also allow for quick retrieval of elements in sorted order.

An example is an OrderedDict in Python, which preserves the order of insertion but does not automatically sort the elements. To have both order and sorting, structures like TreeMap in Java can be used, which stores elements in sorted order by key but does not preserve insertion order.

If a structure is needed that simultaneously preserves insertion order and provides sorting, a combined approach can be implemented: for instance, storing elements in a list for insertion order and in a balanced tree for sorting.

Thus, standard libraries usually have structures either with ordering or with sorting, but to support both properties simultaneously, a custom implementation may be required.