In a small bowl, beat the cream cheese, Dijon mustard and mayonnaise until smooth and spreadable.

Lay a tortilla flat and spread a quarter of the cream cheese mixture edge to edge, leaving a half inch bare at the top for sealing.

Layer a quarter of the ham, two slices of Swiss cheese and a quarter of the spinach over the cream cheese, keeping it inside the border.

Roll the tortilla up tightly from the bottom edge, pressing firmly as you go so there are no air pockets.

Wrap the roll tightly in plastic wrap, twisting the ends like a candy wrapper, and repeat with the remaining tortillas.

Refrigerate the wrapped rolls overnight, or at least 4 hours, so they firm up enough to slice cleanly.

Unwrap and trim off the ragged ends with a sharp knife, then slice each roll into 1-inch rounds.

Arrange the pinwheels cut side up on a platter and keep chilled until serving.