Scroll texture shader on isometric tile cells?

Question

Is it possible to scroll a texture on an isometric tile in a direction (say, Vector(1.0,-1.0) and have the cell look correct? I've tried it with this basic shader applied to a tile:

shader_type canvas_item;

uniform vec2 direction = vec2(0.0,0.0); uniform float speed = 1;

void fragment() { COLOR = texture(TEXTURE, UV + (direction * TIME * speed)); }

But it just causes the entire tile to warp in square shaped blocks, rather than just the isometric shape of the tile. Any way of doing this properly?

Without shader: https://imgur.com/U3DQhAW
With shader: https://imgur.com/YqbJ2eU

Answer 1

Thanks so much to @kleonc for the help, here's the working shader!

shader_type canvas_item;

uniform vec2 direction = vec2(0.0, 0.0);
uniform float speed = 1.0;

vec2 toIsoUV(vec2 uv)
{
    return vec2(uv.x + uv.y - 0.5, -uv.x + uv.y + 0.5);
}

vec2 fromIsoUV(vec2 uv)
{
    return vec2(0.5 * (uv.x - uv.y + 1.0), 0.5 * (uv.x + uv.y));
}

void fragment()
{
    vec2 uv = UV;
    uv -= direction * speed * TIME;
    uv = toIsoUV(uv);
    uv -= floor(uv); // make it into [0, 1) x [0, 1)
    uv = fromIsoUV(uv);
    COLOR = vec4(texture(TEXTURE, uv).rgb, texture(TEXTURE, UV).a);
}