The pieces came apart easily, to my surprise.  The vise bars were held in place with steel dowels that I pressed out with an arbor press.  The hydraulic cylinder worked well, so I left it alone.  A wire wheel removed most of the rust, and in the tightest places, a drill-mounted wire brush handled the rest.  I then used steel wool pads to scrub the surfaces clean.  Once the pieces were down to bare metal, I used lacquer thinner to ensure the surfaces had no remaining soap or oil film.

I selected Rust-Oleum Farm Implement paint mainly because I had a few cans on the shelf.  The original base was black, with the vertical swivel being red.  This time the vise would be totally black.  I masked the areas that were not to be painted, then painted 3 to 4 coats on each piece.

After drying overnight, the pieces went together quickly.  I lubricated the moving parts with a high-quality engine assembly lube, also because I had it on the shelf.  The vise closes smoothly, and the hydraulic arm holds it securely with a twist of the knob.  This vise will look good on the work bench, and should have many years of service left in it.