We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably
less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds
which are symplectically cobordant to something overtwisted, or to the tight 3-sphere, or which admit symplectic caps containing
symplectically embedded spheres with vanishing self-intersection. These constructions imply new and simplified proofs of several
recent results involving fillability, planarity, and non-separating contact type embeddings. The cobordisms are built from symplectic
handles of the form Σ × D and Σ × [−1, 1] × S1, which have symplectic cores and can be attached to contact 3-manifolds
along sufficiently large neighborhoods of transverse links and preLagrangian tori. We also sketch a construction of J-holomorphic
foliations in these cobordisms and formulate a conjecture regarding maps induced on Embedded Contact Homology with twisted
coefficients.