A set of thirty-six chemical elements, anagrammed into a different set of thirty-six elements.

hydrogen + xenon + barium + tantalum + boron + praseodymium + iridium + hassium + plutonium + thallium + germanium + scandium + thulium + einsteinium + erbium + cadmium + beryllium + tin + actinium + seaborgium + carbon + fluorine + indium + osmium + nitrogen + potassium + lead + protactinium + silicon + lutetium + rhenium + mercury + argon + neodymium + platinum + thorium = lanthanum + oxygen + terbium + radon + samarium + dysprosium + iodine + bohrium + aluminium + chromium + palladium + tungsten + lithium + caesium + dubnium + meitnerium + niobium + ytterbium + gallium + arsenic + iron + sodium + nobelium + francium + astatine + strontium + copper + gadolinium + yttrium + selenium + curium + chlorine + promethium + gold + uranium + antimony

Also, if you replace each element with its atomic number (= position in the periodic table), there is still equality.

H + Xe + Ba + Ta + B + Pr + Ir + Hs + Pu + Tl + Ge + Sc + Tm + Es + Er + Cd + Be + Sn + Ac + Sg + C + F + In + Os + N + K + Pb + Pa + Si + Lu + Re + Hg + Ar + Nd + Pt + Th H 1   He 2 Li 3 Be 4   B 5 C 6 N 7 O 8 F 9 Ne 10 Na 11 Mg 12   Al 13 Si 14 P 15 S 16 Cl 17 Ar 18 K 19 Ca 20 Sc 21 Ti 22 V 23 Cr 24 Mn 25 Fe 26 Co 27 Ni 28 Cu 29 Zn 30 Ga 31 Ge 32 As 33 Se 34 Br 35 Kr 36 Rb 37 Sr 38 Y 39 Zr 40 Nb 41 Mo 42 Tc 43 Ru 44 Rh 45 Pd 46 Ag 47 Cd 48 In 49 Sn 50 Sb 51 Te 52 I 53 Xe 54 Cs 55 Ba 56 La* 57 Hf 72 Ta 73 W 74 Re 75 Os 76 Ir 77 Pt 78 Au 79 Hg 80 Tl 81 Pb 82 Bi 83 Po 84 At 85 Rn 86 Fr 87 Ra 88 Ac# 89 Rf 104 Db 105 Sg 106 Bh 107 Hs 108 Mt 109 Ds 110       * Ce 58 Pr 59 Nd 60 Pm 61 Sm 62 Eu 63 Gd 64 Tb 65 Dy 66 Ho 67 Er 68 Tm 69 Yb 70 Lu 71     # Th 90 Pa 91 U 92 Np 93 Pu 94 Am 95 Cm 96 Bk 97 Cf 98 Es 99 Fm 100 Md 101 No 102 Lr 103   1 + 54 + 56 + 73 + 5 + 59 + 77 + 108 + 94 + 81 + 32 + 21 + 69 + 99 + 68 + 48 + 4 + 50 + 89 + 106 + 6 + 9 + 49 + 76 + 7 + 19 + 82 + 91 + 14 + 71 + 75 + 80 + 18 + 60 + 78 + 90   =   La + O + Tb + Rn + Sm + Dy + I + Bh + Al + Cr + Pd + W + Li + Cs + Db + Mt + Nb + Yb + Ga + As + Fe + Na + No + Fr + At + Sr + Cu + Gd + Y + Se + Cm + Cl + Pm + Au + U + Sb H 1   He 2 Li 3 Be 4   B 5 C 6 N 7 O 8 F 9 Ne 10 Na 11 Mg 12   Al 13 Si 14 P 15 S 16 Cl 17 Ar 18 K 19 Ca 20 Sc 21 Ti 22 V 23 Cr 24 Mn 25 Fe 26 Co 27 Ni 28 Cu 29 Zn 30 Ga 31 Ge 32 As 33 Se 34 Br 35 Kr 36 Rb 37 Sr 38 Y 39 Zr 40 Nb 41 Mo 42 Tc 43 Ru 44 Rh 45 Pd 46 Ag 47 Cd 48 In 49 Sn 50 Sb 51 Te 52 I 53 Xe 54 Cs 55 Ba 56 La* 57 Hf 72 Ta 73 W 74 Re 75 Os 76 Ir 77 Pt 78 Au 79 Hg 80 Tl 81 Pb 82 Bi 83 Po 84 At 85 Rn 86 Fr 87 Ra 88 Ac# 89 Rf 104 Db 105 Sg 106 Bh 107 Hs 108 Mt 109 Ds 110       * Ce 58 Pr 59 Nd 60 Pm 61 Sm 62 Eu 63 Gd 64 Tb 65 Dy 66 Ho 67 Er 68 Tm 69 Yb 70 Lu 71     # Th 90 Pa 91 U 92 Np 93 Pu 94 Am 95 Cm 96 Bk 97 Cf 98 Es 99 Fm 100 Md 101 No 102 Lr 103   57 + 8 + 65 + 86 + 62 + 66 + 53 + 107 + 13 + 24 + 46 + 74 + 3 + 55 + 105 + 109 + 41 + 70 + 31 + 33 + 26 + 11 + 102 + 87 + 85 + 38 + 29 + 64 + 39 + 34 + 96 + 17 + 61 + 79 + 92 + 51

[= 2019]

Furthermore, if you take the letter values (A=1, B=2, C=3 etc.) of each element abbreviation (e.g. lanthanum (La) = 12+1 = 13)
and add them up, the sum of each side is the same - yet another level of equality, making this a "triply-true" anagram.

H + Xe + Ba + Ta + B + Pr + Ir + Hs + Pu + Tl + Ge + Sc + Tm + Es + Er + Cd + Be + Sn + Ac + Sg + C + F + In + Os + N + K + Pb + Pa + Si + Lu + Re + Hg + Ar + Nd + Pt + Th	8 + 29 + 3 + 21 + 2 + 34 + 27 + 27 + 37 + 32 + 12 + 22 + 33 + 24 + 23 + 7 + 7 + 33 + 4 + 26 + 3 + 6 + 23 + 34 + 14 + 11 + 18 + 17 + 28 + 33 + 23 + 15 + 19 + 18 + 36 + 28
=
La + O + Tb + Rn + Sm + Dy + I + Bh + Al + Cr + Pd + W + Li + Cs + Db + Mt + Nb + Yb + Ga + As + Fe + Na + No + Fr + At + Sr + Cu + Gd + Y + Se + Cm + Cl + Pm + Au + U + Sb	13 + 15 + 22 + 32 + 32 + 29 + 9 + 10 + 13 + 21 + 20 + 23 + 21 + 22 + 6 + 33 + 16 + 27 + 8 + 20 + 11 + 15 + 29 + 24 + 21 + 37 + 24 + 11 + 25 + 24 + 16 + 15 + 29 + 22 + 21 + 21

[= 737]

NOTES:

(1) I spent some time thinking about the conflicting spellings which exist for three elements: aluminium/aluminum, caesium/cesium, and sulphur/sulfur. I decided that the "right" thing to do was to use the IUPAC-standardized names, a list of which can be found on the IUPAC web site.
Note that Caesium is listed as the preferred spelling for that element, with cesium as a (presumably) acceptable alternative, and the same for Aluminium/aluminum. But note that Sulfur (yes, the "American" spelling!) is the only spelling listed for that element. From this I conclude that the "right" spellings to use for these three elements is Caesium, Aluminium, and Sulfur. As it turns out, I don't use Sulfur above, just the other two. At any rate, the anagram uses only the IUPAC-preferred spellings for all elements, which I think makes it an improvement over all my previous chemical anagrams.

(2) A side note for those who might be interested: element 110 was given a name by IUPAC (Darmstadtium (Ds), after Darmstadt, Germany where it was discovered) only very recently, in August 2003. I allowed inclusion of this element if need be, but I didn't end up using it.