Is a square tube more resistant to bending than a round tube? In considering tubular forms for aircraft construction, I am reasoning that a square form (or I-beam) would be more resistant to bending (if the load is directly perpendicular and in the plane of the I, for example how I beams are set in bridges to support against gravity), while a round tube would be more resistant to torsional (twisting) and compressive loads.
The comparison would made for the same weight and roughly the same dimensions (throw the I beam in there too if you would like).
Is this reasoning correct? Are there structures that combine the virtues of the two?
 A: You need to be a bit more specific to make claims like that.  Obviously to say a square tube is more resistant to bending than a steel tube, it would depend on the properties of the tubes, such as the sizes and materials.
The way I've typically seen them compared is to look at the strength to weight ratio when using the same material.  Strength to weight ratio is very important for aircraft; which is also why they use a lot of materials like aluminum and titanium. 
Because of the way bending moments are applied, you want material where it can provide the most resistance against the load.  For bending, that means you need material in the axis of movement; and the further away from the neutral axis, the more strength it can provide for the weight (though depending on thickness, this may introduce buckling if you make them too far from the axis).  I-beams are a really good example; they have a lot of material in the bending direction, with both the flanges spaced away from the neutral axis.
Since a round bar doesn't really focus the material far away from the neutral axis of bending, a square tube will give you a better strength to weight ratio against bending.  A round tube will be the same for torsion.
As far as resisting both goes, a hollow steel tube wouldn't be a bad start.  It's a bit of a mix between I-beam and round tube.  I'm sure you could use some sort of optimization to get a better shape, or even tailor the shape to the expected loads if you really wanted to optimize the shape.
A: Bending stress and torsional stress are both inversely proportional to the moment of inertia (second moment of area) of the cross section. For torsional stress the polar moment of inertial is normally used. For bending stress it is the centroidal moment of inertia. 
You can find the equations for both types of moments of inertia for a square and circular cross section in any statics textbook, or on-line. You can then compare them to determine which cross section is more resistant to torsional vs bending stress. 
Of course to determine the actual bending stress and torsional stress of each, you need equations for these as well, and the applicable material properties as well.
Hope this helps.
