Why alkanes containing even number of carbons can be packed more efficiently than alkanes with odd number of carbon atoms?
A recent article in PubMed (Angew Chem Int Ed Engl. 2016 Nov 2;55(45):14090-14095.) suggests that this odd-even effect also affects translation diffusion in the liquid state. 

In the introduction of the PubMed article, it also addresses your particular question and confirms your intuition that it relates to the symmetry of the molecule. Interestingly, this was only experimentally confirmed in the last decade. To quote the article: 


  Although this striking odd–even effect was known hundreds of years ago, its molecular origin was only revealed in the last decade. With the help of single-crystal diffraction and controlled crystal growth, the crystalline lattice of n-alkanes was determined.[5] It was confirmed that the intermolecular distances between the end groups, namely, the CH3 groups, are responsible for the alternation in the densities and melting points. In short, n-alkanes with an even number of carbon atoms pack more efficiently into ordered periodic crystalline structures and thus exhibit higher densities and melt at higher temperatures, whereas n-alkanes with an odd number of carbon atoms do not pack as effectively, and thus their densities are lower, and lower temperatures are needed for melting or freezing. A “packing effect” has been established to explain the odd–even effect as well as the related physical properties in the solid state. 


Edit: For completeness, this is the article they link to that originally proved the effect experimentally: http://onlinelibrary.wiley.com/doi/10.1002/(SICI)1521-3773(19990401)38:7%3C988::AID-ANIE988%3E3.0.CO%3b2-0/full