Can I apply unique colour, shape or size attributes to the data points in each classification polygon and then save them to a new variable column so I can view the classification as text?

A: Yes.  When you have a diagram open use the icons on the RHS toolbar such as Colour Rows by Polygon, Shape Rows by Polygon, Size Rows by Polygon to attribute the data.  To save the attributes as new columsn in the dataset select the appropriate Make Variable From… option from the Data menu.