This is the first of two papers devoted to the local "metric-like" unconstrained Lagrangians and field equations for higher-spin gauge fields of mixed symmetry in flat space. Here we complete the previous constrained formulation of Labastida for Bose fields. We thus recover his Lagrangians via the Bianchi identities, before extending them to their "minimal" unconstrained form with higher derivatives of the compensator fields and to yet another, non-minimal, form with only two-derivative terms. We also identify classes of these systems that are invariant under Weyl-like symmetries.

83 pages, LaTeX. References and comments added, typos corrected. Final version to appear in Nuclear Physics B