Hi Michael
My grandfather used hardboard for all of his wood projects and I recall him saying always glue the smooth side down, it will hold unlike the checker board side.
I do use it for drawers because it's strong and true and flat.
But I don't use it to offen because it will lift in time because the glue can't sink into the wood fibers to get a good bond.
I do see members use it on router tables alot and rap it with oak banding to cover the open look of the hardboard.
He always used contack glue, I recall he had it on all of his work shop benchs with many,many saw marks on it... and slotted wood screws to hold it down in place
it's funny what you recall when you and I worked in the shop with our grandfathers. 
I do recommend 1/4" MDF because the yellow glue will bond to it and you can keep it looking like new with just a little TLC and some Johnson and Johnson floor wax and it's slick and smooth......
Bj
My grandfather used hardboard for all of his wood projects and I recall him saying always glue the smooth side down, it will hold unlike the checker board side.
I do use it for drawers because it's strong and true and flat.
But I don't use it to offen because it will lift in time because the glue can't sink into the wood fibers to get a good bond.
I do see members use it on router tables alot and rap it with oak banding to cover the open look of the hardboard.
He always used contack glue, I recall he had it on all of his work shop benchs with many,many saw marks on it... and slotted wood screws to hold it down in place
I do recommend 1/4" MDF because the yellow glue will bond to it and you can keep it looking like new with just a little TLC and some Johnson and Johnson floor wax and it's slick and smooth......
Bj