We investigate $\sqcap$-directoids which are bounded and equipped by a unary operation which is an antitone involution. Hence, a new operation $\bigsqcup$ can be introduced via De Morgan laws. Basic properties of these algebras are established. On every such an algebra a ring-like structure can be derived whose axioms are similar to that of a generalized boolean quasiring. We introduce a concept of symmetrical difference and prove its basic properties. Finally, we study conditions of direct decomposability of directoids...