Mechanical properties of biological molecular aggregates are essential to their function. A remarkable example are double-stranded DNA viruses such as the φ29 bacteriophage, that not only has to withstand pressures of tens of atmospheres exerted by the confined DNA, but also uses this stored elastic energy during DNA translocation into the host. Here we show that empty prolated φ29 bacteriophage proheads exhibit an intriguing anisotropic stiffness which behaves counterintuitively different from standard continuum elasticity predictions. By using atomic force microscopy, we find that the φ29 shells are approximately two-times stiffer along the short than along the long axis. This result can be attributed to the existence of a residual stress, a hypothesis that we confirm by coarse-grained simulations. This built-in stress of the virus prohead could be a strategy to provide extra mechanical strength to withstand the DNA compaction during and after packing and a variety of extracellular conditions, such as osmotic shocks or dehydration.