Warren in Lancaster, PA
This is a skew rabbet plane. We don't use rabbet planes for making grooves. We use them for making rabbets, adjusting rabbets, and making fillets in mouldings. You can use it for trimming a tenon, but I would be leery about using it to clean off glue because the glue would be hard on the plane sole.
This type of rabbet plane is used for deepening rabbets where the right side of the plane rubs against the shoulder of the rabbet. (A square mouth rabbet can be used left handed also, where the left side rubs against the stock, but a skew rabbet cannot.) The iron is from a different plane. On the right side of the plane, looking from the top, the iron has to protrude a small amount for clearance. You want it so that a straight edge held against the side of the plane will rock a little on the edge of the iron. The plane will work the way it is but better if the iron doesn't protrude quite so much.
On the left side of the plane, the iron does not have to match up at all. It can be short of the edge of the beech, or long the way it stands now.