What do you consider "serious saving"? If this means eating ramen for months, or otherwise limiting even remotely important things then don't do it. You are enjoying your music and the heavens won't open when using the HD800(s) (I use a HD800, triangulating in regard to the HD560s based on my experience with other headphones that others rate about the same).

Having said that, if you can afford to save without it messing with your life, the HD800 (+ EQ) may well be worth it. There is lots of music where I'm absolutely fine with my work headphones (SHP 9600, because comfort), my out-and-about Galaxy Buds or the trying-out-the-current-low-end Moondrop Chu or Salnotes Zero. But for classical, especially symphonic, music I pretty much always use the best headphones/IEMs available to me and usable in the situation (so most of the time the HD800) because it helps my enjoyment. The technical improvements such as better instrument separation and placement make it easier to understand what is happening and to just get lost in the music.