Labelling and drawing at rotated/shifted objects

I'm trying to draw an image from a mathematical publication (see the crop at the end of my post). You can find a recent version of the drawing at page 37 of my script (PDF). The complete source is here.

I drew a rhomb with those squares in it and shifted all pieces to the right place:

\begin{tikzpicture}
\draw (0,0) -- (4,1) -- (8,0) -- (4,-1) -- (0,0);
\draw[thick] (6,.5) -- (4,1) -- (4,-1) -- (6,-.5)
[yshift=4cm]  (6,.5) -- (4,1) -- (4,-1) -- (6,-.5);
\draw[thick,dashed] (6,.5) -- (6,-.5)
[yshift=4cm]  (6,.5) -- (6,-.5);

I did the same with the squares around those figures. Now I want to label each square (R_0, R_1) and draw arrows from one box to another.

I'm looking for a better way than to write something like \draw[->] (2,2) -- (2,0);. So adding nodes sounds like the right idea, but I found no working solution. What would you recommend for drawing arrows and adding some labels?

-

You can use the shapes tikz-library. I am including an example, but you should read section 48.3 Geometric shapes on page 420 from the tikz manual. Here is the example

\documentclass{minimal}

\usepackage{tikz}
\usetikzlibrary{shapes}
\usetikzlibrary{positioning}

\begin{document}

\begin{tikzpicture}

\node[draw=lightgray,rectangle,inner sep=0.5cm,outer sep=5pt,label=245:$R_0$] (top) {\tikz{
\node[draw,trapezium,trapezium angle=75,shape border rotate=270,outer sep=0pt] (top trap)
{$\sigma(S_0)$};
\draw[white,postaction={draw=black,dashed}]
(top trap.bottom left corner) -- (top trap.bottom right corner);
}};

\node[draw=lightgray,rectangle,inner sep=0.5cm,below=2cm of top,outer sep=5pt,label=245:$R_0$] (bottom) {\tikz{
\node[draw,trapezium,trapezium angle=75,shape border rotate=270,outer sep=0pt] (trap)
{$T_0$};
\draw[white,postaction={draw=black,dashed}] (trap.top left corner) -- (trap.top right corner);
}};

\draw[->] (top) -- (bottom) node[pos=0.5,right] {$\varphi|_{R_0}$};
\end{tikzpicture}

\end{document}

The result is

-