This invention relates to tiers of kerf engaging elements at saw blade sides. The tiers recede from the blade's most forward peripheral cutting elements and project laterally to engage the kerf in incremental steps as the blade advances into the material addressed. Each tier is designed to perform a distinctive action. Individual tiers may be designed to plane, or shave, or scratch, or scrape, or sand, or burnish. The mentioned preferred embodiment of this invention submits the kerf to coarse sanding, fine sanding, and burnishing with one pass of the blade.