Just after I asked question here (so before I got any answers on TeX.SE) micket from #latex@irc.freenode.net (where I asked similar question, but only about centering (0,0)) said that I can just put tikzpicture within a node being in the center of the page. I must admit I overlooked this embedding feature of TikZ.
Simple as that, but apparently not as simple as it seems. After all early versions of Altermundus's answer were exploiting this technique, but with some quirk. This answer is to provide a way of embedding tikzpicture in a node of another tikzpicture without setting shift key.
Example
\documentclass{minimal}
\usepackage[a6paper,margin=0pt]{geometry}
\usepackage{tikz}
\begin{document}
\pagestyle{empty}
\parindent=0pt
\begin{tikzpicture}[remember picture,overlay,x=25mm,y=25mm]
\draw[lightgray]
(current page.north) -- (current page.south)
(current page.west) -- (current page.east);
\draw
(current page.center) circle (5pt);
\node at (current page.center) {%
\begin{tikzpicture}[overlay]
\draw
(-1,-1) -- (0,1) -- (1,-1) -- (2,1);
\end{tikzpicture}%
};
\end{tikzpicture}
\end{document}

Now you may ask why anyone would like to use such technique, which isn't as concise as tikzpicture's shift. I'll tell you, because this is why I ended using this technique.
Remark
Above technique gives you possibility of easily changing the point of inner tikzpicture which is being positioned (according to the node where it is put):
- from
tikpicture's (0,0)
- to
tikzpicture's center.
How? By removing overlay from inner tikzpicture options. Lack of it was the reason behind Altermundus' quirk with shift in the outer tikzpicture node. He was so close...

Afterword
It's true that remark has nothing to do with original question, yet it was important in my case, thus I thought I should share it with you as a separate answer. Hope it will be useful for others.