Your first guess was probably your best guess...it's about balance.  This used be common on heavy workboat oars, sometimes in conjunction with oars with holes in the square part meant to fit over single, fixed thole pins.  Some folks like the way the square inboard section brings the balance closer to neutral.  Personally,  I row mostly with my fingers hooked around the grips rather then holding it all the way up in the crotch between index finger and thumb, so I like lighter oars all around.  Everybody has their own preferences that way.

.....Michael

   Balance. But the square-shaped section also saves considerable work when making the oars. Well-balanced oars will save the oarsman a lot of energy when rowing. To achieve that balance, I drilled a hole in the end of each oar and filled it with lead.

Do the square shafts keep the oars from slipping overboard through round oarlocks?

Here's John's video showing him rowing with such a pair, which ought to explain better'n I could:

https://www.youtube.com/watch?v=snsfkwWIYdE

He's using horn-type oarlocks there.  Round oarlocks likely wouldn't fit over the square part.

CLC's oar plans, templates, etc., are here:

https://www.clcboats.com/shop/boats/boat-gear/oars/wooden-oars-plans-patterns.html

.....Michael

   I slightly rounded the square edges of my oars so that they (just barely) slide through the hole of round oarlocks. This makes them quite useful and also much less likely to slip out of the oarlocks.

Thanks all for the info.  One last question:  Some oars' inboard ends (i.e., the hand grips) do not overlap and some inboard ends overlap and must be physically separated by the rower during the stroke.  What are the pros/cons of each of these configurations?  Personal preference?  Greater leverage?

A couple inches of overlap on the recovery is normal for typical fixed-seat, recreational rowboats.  It's not difficult to manage without hurtin' yourself, with a bit of practice.

In most cases, having that overlap on recovery will place the hands at a comfortable shoulder width and a bit below shoulder height on the power stroke, if everythiung is happily proportioned, allowing the oarsman to get a good, strong pull.  Having the hands too close together, too far apart, too high, or too low, will all rob the oarsman of a good portion of the slim fraction of a horsepower available,

There are a lot of variables here: seat height relative to the oarlocks and oarlock height relative to the water, distance from the seat to the oarlocks, spread of the oarlocks, length of the oars, and the location and nature of the point where the oars bear on the oarlocks...all factor into it, which is why a good rowboat is hard to design and why a good rowboat which is also a good sailboat is especially difficult to design.  If things don't work out happily, she'll sail like a bathtub with bedsheets on broomsticks, row like a horse trough on a sea of molasses, and track as if being steered by a drunken baboon.

Sometimes concessions and tradeoffs have to be made.  For example, a beamy boat (which might make a good sailboat) might require impossibly long oars with impractical geometry for best rowing, so you settle for shorter oars with a bit more spread at the hands than would be ideal, and you try to arrange a rowing thwart a bit higher to compensate somewhat for the increased angle (and thus grip height) needed to get the blades in the water.  If she's primarily to be a sailing boat, that's an acceptable compromise.  A slender boat, on the other hand, would pose a whole different set of compromises to be made.

Tricky stuff, boat design.  The trick lies in making good tradeoffs which will make the boat better for her primary purpose without making her useless for secondary purposes.

.....Michael