Motivated by the computation of some Feynman amplitudes, Broadhurst and Roberts recently conjectured and checked numerically to high precision a set of remarkable quadratic relations between the Bessel moments ∫∞0I0(t)iK0(t)k−it2j−1dt(i,j=1,…,⌊(k−1)/2⌋), where k≥1 is a fixed integer and I0 and K0 denote the modified Bessel functions. We interpret these integrals and variants thereof as coefficients of the period pairing between middle de Rham cohomology and twisted homology of symmetric powers of the Kloosterman connection. Building on the general framework developed by Fresan, Sabbah and Yu (2020), this enables us to prove quadratic relations of the form suggested by Broadhurst and Roberts, which conjecturally comprise all algebraic relations between these numbers. We also make Deligne’s conjecture explicit, thus explaining many evaluations of critical values of L-functions of symmetric power moments of Kloosterman sums in terms of determinants of Bessel moments.