There are many ways in which Beethoven can be seen as both a classical and romantic composer.
He is seen as classical in many respects. A quote from Charles Rosen, The Classical style reads
"Beethoven enlarged the limits of the classical style beyond the previous conceptions, but never changed its essential structure or abandoned it, as did the composers who followed him. In the other fundamental aspects of his musical language, as well as in the key relations within the single movement, Beethoven may be said to have remained within the classical framework,even while using it in radical and original ways"
His symphony no 6, Pastoral, is one of the main pieces that bridges many of the gaps between the classical and romantic era. It is classical in that it is rooted in a tonic-dominant relationship, and includes fugue, cannon and theme and variation.
However it can be seen as romantic in that it takes a lot of influence from natural and the natural world. Beethoven was coached and definitely influenced by Haydn, a renowned nature lover for a year. In the Classical period, music did not tend to pertain to this type of influence. The Pastoral symphony however is based largely around nature. It is also programmatic, a distinctive element of romantic writing.
By 1808, the time of the pastoral symphony, the romantic movement in poetry was in full swing. Instead of the standard classical 4 movements, the pastoral has 5.
The 9th symphony seems to be the main piece of writing that crosses the bridge between the classical and romantic periods, and is seen as the missing link bringing the classical period fully to an end.