This paper is a continuation of the investigation of the properties of invertible disjointness preserving operators (d.p.o) undertaken by the authors. A disjointness preserving operator is a linear operator \(T : X \rightarrow Y\), where \(X\) and \(Y\) are vector lattices, with the property that if \(x_1 \perp x_2\) then \(T x_1 \perp T x_2\). Linear operators are called \textit{positive} if they preserve inequalities and are called \textit{regular} if they can be written as the difference of two positive linear operators. This paper is concerned with the following questions. Problem A: Let \(T : X \rightarrow Y\) be an injective d.p.o. Under what additional conditions on \(X, Y\) and \(T\) is the operator \(T^{-1} : TX \rightarrow X\) also disjointness preserving? Problem B: Under what conditions on \(X, Y\) and disjointness preserving \(T: X \rightarrow Y\) are the vector lattices \(X\) and \(Y\) order isomorphic? Problem C: Under what conditions on \(X\), \(Y\) and \(T\) is the operator \(T\) regular? All these problems have negative solutions, as was shown by the authors in earlier works. On the other hand, under very general conditions these problems have affirmative solutions, most of which are proved or reproduced in this paper. The purpose of the present paper is to solve the preceding three basic problems for the most common classes of vector lattices. The paper has eight sections. The first two consist of an introduction, definitions, notation and some auxilary results. In Section 3, vector lattices, for which any d.p.o from them to any other vector lattice being regular is characterized. In Theorem 3.2.1, a class of vector lattices \(X\) such that for any injective d.p.o, \(T : X \rightarrow Y\), the inverse \(T^{-1} : TX \rightarrow Y\) is also d.p.o, is considered. A necessary and a sufficient condition is given for a vector lattice to be in this class. Definition: A vector lattice \(X\) is called \textit{\(d\)-rigid} if for any vector lattice \(Y\) and for any bijective d.p.o \(T : X \rightarrow Y\), \(T^{-1}\) is also d.p.o. A vector lattice \(X\) is called \textit{super \(d\)-rigid} if any bijective operator \(T\) from \(X\) onto an arbitrary vector lattice \(Y\) is regular. In Section 4, the authors describe a large class of \(d\)-rigid and super \(d\)-rigid domains. In Section 5, \(d\)-rigidness, super \(d\)-rigidness of the domains of d.p.o, as well as regularity of an \(d\)-isomorphism are investigated. The class of \textit{weakly \(c_0\)-complete} vector lattices is introduced and for this large class of vector lattices serving as domains, conditions under which every bijective d.p.o is a \(d\)-isomorphism and a condition under which every \(d\)-isomorphism is regular are given. In particular, complete answers to problems A--C are obtained when the domain \(X\) is relatively uniformly complete and the range \(Y\) has the countable sup property. A well-known theorem due to Huijmans--de Pagter--Koldunov (HPK theorem) states that if \(X\) is an \(r_u\)-complete vector lattice, \(Y\) is a normed vector lattice and \(T: X \rightarrow Y\) is an injective d.p.o, then \(x \perp z\) iff \(T x \perp Tz\). In Section 7, the authors study the following questions: (1) To what extent can condition \(s\) on \(X\) and \(Y\) in the HPK theorem be weakened? (2) Under what conditions can we interchange \(X\) and \(Y\) in the HPK theorem? In the last section, previously obtained results are applied to vector lattices of continuous functions serving as domain, range or both for a disjointness preserving operator. We note that this interesting paper also contains some open problems in the area.