In the paper "Local Index Formula in Noncommutative Geometry" Connes and Moscovici build the spectral triple $(A,H,D)$ where $A=C^{\infty}_c(P) \rtimes \Gamma$ where $\Gamma$ is an arbitrary subgroup of orientation preserving diffeomorphism of $M$ and $P=GL^+(M)/SO(n)$ is the bundle of euclidean metric. Some more details of the construction can be found in this question. The relevant operator $D$ is defined by the equation $D|D|=Q$ where $Q=(d_Vd_V^*-d_V^*d_V) \oplus (d_H+d_H^*)$ (see also the above link).
What is the reason for the vertical part to be second order while horizontal part is first order?
The authors argue that (in general, for even dimensional (Riemannian) manifold $M$) the operator $dd^*-d^*d$ represents the signature class of $M$ by showing that $d+d^*$ is homotopic to $\Delta^{-\frac12}(dd^*-d^*d)$ (here $\Delta=dd^*+d^*d$) which is fine but here already I have some doubts: it is tempting to say that this is homotopic with $dd^*-d^*d$ via $$(\Delta^{-\frac12+\frac{t}{2}}(dd^*-d^*d))_{t \in [0,1]}.$$ However I'm only familiar with the construction of $K$-homology class for the differential operator of the first order as explained in "Analytic K-homology" by Higson and Roe. But okay, even if we assume that $dd^*-d^*d$ represents the signature class of $M$ it justifies the name signature operator for $Q$ but it does not explain why the vertical component is taken to be second order and the horizontal to be first order. In the pseudodifferential calculus explained in "Local Index Formula" paper the degree of the differential operator is defined in such a way that $d_H+d_H^*$ becomes second order but I don't see the good conceptual reason for this.
