Not only can you bathe birds, but it's essential to their health and happiness! They need to have their feathers clean and in good condition, and most wild birds will bathe daily in water, or if there is no water available will have a dust bath.
They absolutely love the opportunity to bathe, the best way to do it is if you set the spray to fine droplets and aim it above them so it falls on them like rain. They do indeed do a 'shower dance', where they spread their wings out to catch the drops, raise all their feathers up so that the water soaks through them, and shake and shimmy up and down their perch. Then they will hang upside down to ensure a full soaking on each side. They usually scream and shriek with excitement and many birds will get very excited about the sound of running water.
I don't bathe Jasper much in the winter because there are a few draughts in the flat and a wet bird in a draught can very quickly be a dead bird, but in summer I usually spray him every few days.