Why do polysaccharides consisting of alpha-glucose have helical structures?
Note: This is an off the top of my head answer (I might be horribly wrong), but it has prompted me to do some research. I will add more details, and/or make corrections once I am done. In the meantime, I would love to hear what others have to say on this.

Pictured below is the 2D structure of amylose



This primarily composed of $\alpha$-(1-to-4)-D-glucose units. Although the $\alpha$-(1-to-4) links are capable of relatively free rotation about their torsional angles, hydrogen bonding between the $\ce{O}$3 and $\ce{O}$2 encourages helicogenesis. 

Single helical amylose has hydrogen-bonding between $\ce{O}$2 and $\ce{O}$6 atoms on outside surface of the helix and the ring oxygen pointing inwards. 

Contrat this with cellulose which is a linear polymer of $\beta$-(1-to-4)-D-glucopyranose units. 

The fully equatorial conformation of $\beta$-linked glucopyranose residues stabilises the chair structure, minimising its flexibility. 

Pictured below, the 2D structure of cellulose and displaying the intrastrand hydrogen bonding 



Interactive 3D visualisation for amylose and cellulose.