We consider the dynamical model of a binary bosonic gas trapped in a symmetric dual-core cigar-shaped potential. The setting is modeled by a system of linearly coupled one-dimensional Gross-Pitaevskii equations with cubic self-repulsive terms and quadratic attractive ones, which represent the Lee-Huang-Yang corrections [T. D. Lee, K. S. Huang, and C. N. Yang, Phys. Rev. 106, 1135 (1957).PHRVAO0031-899X10.1103/PhysRev.106.1135] to the mean-field theory in this geometry. The main subject is spontaneous symmetry breaking (SSB) of quantum droplets (QDs), followed by restoration of the symmetry, with respect to the identical parallel-coupled trapping cores, following the increase of the QDs' total norm. The SSB transition and inverse symmetry-restoring transition form a bifurcation loop, whose shape is concave at small values of the intercore coupling constant κ and convex at larger κ. The loop does not exist above a critical value of κ. At very large values of the norm, QDs do not break their symmetry, featuring a flat-top shape. Some results are obtained in an analytical form, including an exact front solution connecting asymptotically constant zero and finite values of the wave function. Collisions between moving QDs are considered too, demonstrating a trend to merge into breathers.