This cigar cutter functions by means of 3 blades that float on springs. The springs keep the blades up against the underside of the cutting guides (cutter head housing). Despite 4 holes, there are actually only 3 cutters: Two separate "V" cutters, and a single flat cutter that makes both straight cuts, one small and one large. The V-cutters are the same size; the cut they make is governed by the thickness of the top piece. Dismantling and sharpening these cutters can be tricky. Here's what I did.
Disassembly: From the underside, I removed the large rectangular plastic plug with your fingers. Next came the 4 long mounting bolts that hold the cutter head housing onto the chassis by using a #2 Phillips; see image 1. I turned the cutter right-side up and lifted off the cutter head housing. This is when I realized springs are gently pushing up on the cutters to keep them flush with the underside of the cutter head housing. Image 2 shows the cutter head housing off and one side of the cutter assembly has popped out and the springs have fallen into the chassis. NOTE: The entire cutter has 5 springs: 2 lightweight springs and 4 regular springs (image 4), and one large spring for the center axle (images 3 (before cleaning) and 5 (after cleaning)). The main blade is held to the middle 2 pins with tiny screws; be extra careful with these. Anyway, I kept going until everything was apart; image 5. (It wasn't necessary to remove the chassis bushings and spring clips, but what the heck!)
Cleaning: I removed the corrosion off the aluminum cutter head housing and chassis with steel wool. Tarnish on the blades got the Dremel wire brush treatment. All non-cutting parts got sprayed with Tri-Flow then wiped. All cutting parts were wiped liberally with Camillia oil. The plastic snap-in cover on the bottom got dosed with ArmorAll. Finally, during assembly, each pin got a drop of 3-in-1 oil for good measure.
Sharpening: I sharpened the main flat blade free-hand using 600 and 1000 grit Norton water stones, concentrating on the bevel side mostly, and then flattening the back with a couple strokes at the 1000-grit level; it's the same technique as for sharpening plane blades and chisels. The two "V" cutters were a little more problematic. I used a 600-grit lap-style diamond sharpening paddle (normally used for sharpening router bits) to smooth the sides of each cutter's "beak," always stroking from the beak to the front. The "beaks" are curved, so you must free-hand it, slightly rotating the sharpening paddle hone as needed. The goal is to get a nice even burr on the inside portion of each "V" beak. (Until you take the thing apart yourself, none of this makes sense!) I then used a scrap of 800-grit sandpaper to remove this burr from the inside of the "V". This was the most complicated thing I've ever tried to sharpen, but it worked out well. It's definitely sharper than before and cuts like a dream.
Reassembly: I reattached the bushings in the chassis with the spring clips. I threaded the main axle through the chassis, adding the spring as shown in image 6. I attach the main spring retaining screw in place. I screwed the handle to the axle and pulled it forward, rotating the axle so the axle holes are facing up, and dropped in the two outermost pins. I then clamped the handle in this position so the axle holes were facing straight up. I added regular springs to these outermost pins, then assembled everything else as in image 4. I got all 10 fingers around it and dropped it into place before bouncing everything up and down on the springs to ensure all was good; see image 7 and note how the straight cutter's bevel is up. (That's how it was originally, but in hindsight, the bevel should probably be down.) Finally, I put the cutter head housing back on, made sure it was seated nicely, flipped the whole thing over, and put the long bolts back in before snapping the plastic cover back into place. See image 8 for the money shot.
It works REALLY well now, but I won't be leaving it outside on the back patio any longer.