How does specularPower work? I'm confused

I have a box with a wood texture. I’m lighting with a HemisphericalLight.

But, it’s too dark from the side angles. It only looks bright when viewed from the very top.

When I set specularPower to 0, the box looks much brighter, from all angles. Why is this happening? It seems to me that it should be the other way around, that increasing specularPower should increase highlights.

The playground:

Increasing spec power is increasing the sharpness of the highlights so the more spec, the smaller the highlights.

The diffuse contribution stays totally unchanged and is added to the spec contribution.

Hemispherical adds a bit of light in all direction from bottom to top color so considering black on the ground and white on the top the sides would be gray which seems totally accurate.

A nice way to look at lighting impact is to use a sphere: