This question could be answered fairly quickly, if it was not because of semantics. Logically it makes sense to use h1 before h2 in the HTML4 standard, but strangely the standard itself dose not require you to use the heading elements in any specific order.
The heading elements are however not meant to be thrown around on a page, in whatever order that amuses the designer. The headings are serious elements, meant to be used in a certain order. The div element can be used for such experimenting if you prefer, but do try to keep the order of the heading elements logical when using an old standard like HTML4.
Only the Living Standard, aka the HTML5 standard allows the use of H1 exclusively, when used in combination with the new sectioning elements.
What about SEO and ordering of headings?
Some people are claiming that you should use h1 as the first heading, since it carries the most importance if measured in weight. This may be somewhat true, but even if true, it could be said that the value of any one factor has diminished over time, and it is NOT the first place you should look to optimize your site in 2014.
The ordering of the headings on your page contribute little, if at all, to how well the individual page of your site is ranking. It is likely more relevant to look into the overall authority of your site, and the Keywords and Keyphrases found throughout individual pages of your site.
Do not panic over little things like this
Obviously it can be said that using a illogical order of headings is just laziness or sloppiness from the developer or designers part. But you rarely need to go back and re-design your entire site, that is unless it is causing accessibility issues to your users.
Modern search engines like Google, is generally good at ranking your pages, regardless of how many validation errors and how illogical your site was designed, but this dose not mean that you should be a slob. It is important to strive for code validity, mostly for your own sake, but sometimes it can actually also matter to your rankings.
- Does the ordering of heading tags matter? – Matt Cutts on ordering of headings