An "oriented rectilinear k-simplex" is a special case of an "oriented affine k-simplex" in that it requires that the boundaries be straight lines, planes, etc. in a given coordinate system which an oriented affine k-simplex does not.
Are you clear on the distinction between and ? Unfortunately you didn't include the "(75)" where was given but I think it is simply a simplex. , we are told, "is obtained from by interchanging and ". That is, is exactly the same point set as with possibly a different orientation. Being "in E" is a property of the point set, regardless of orientation.