The classification of order types

The classification of order types

In response to a previous question which I asked concerning the order type
of the Rationals, a reference was made (by MDJ) to a theorem of Cantor
stating that any two countable, dense, linearly ordered sets without
endpoints are isomorphic as linear orderings.
My question is : If we remove the endpoints condition, say by looking at
the intersection of Q and [0,1], then is this order type also unique (upto
isomorphism) amongst such linear, dense, countable orderings.
More generally, have all of the (linear) order types which do not
correspond to an ordinal number been classified?