As I understand it, coordinate axes are defined by the tangents to coordinate curves. The coordinate curves are formed by the intersection of coordinate surfaces. The nature of the particular coordinate surfaces will determine if the coordinate axis change direction from point to point.
In a book on general relativity, the author has two statements where basis vectors and basis one-forms are introduced, viz., "basis vectors are tangent to the coordinate curves (along which only one of the coordinates changes), and basis one-forms are gradients of the coordinate surfaces (on which only one of the coordinates remains constant)."
It is the parenthetical statements in bold about which I am puzzled. Specifically, unless all the coordinate surfaces meet orthogonally everywhere, the meeting of which forms the coordinate curves, a restriction not specified in the book, then any movement, even infinitesimal, made along the tangent coordinate axis could change one or more of the other coordinates as well. That is, aren't the bold statements either assuming orthogonality or are overly general?