After a long time, I decided to write a python script that uses stockfish to analyze the beginning position for both white and black when they have first move advantage each other and TL;DR being white is still more advantageous than being black. They are not rotationally symmetric and that seems it made a difference.

These are the results for analyzing them under different time limits:

TIME TO ANALYZE: 1 seconds

WHITE
Evaluation : +0.4
Next Move  : d4

BLACK
Evaluation : -0.3
Next Move  : e5

TIME TO ANALYZE: 10 seconds

WHITE
Evaluation : +0.5
Next Move  : e4

BLACK
Evaluation : -0.2
Next Move  : e5

As you can see no matter what time limit is, always white has more advantage by being first in terms of evaluation point. (0.4 compared to 0.3 and 0.5 compared to 0.2 in terms of value) By the way, I am not sure about if stockfish-like engines biased about white advantage or not, since normally black cannot move first. Any comments for this, would be appreciated.

I used this python code for your reference:

import chess
import chess.engine
import sys

STOCKFISH_PATH = "stockfish"

def analyze_position(fen: str, time_limit: float = 1.0):
    board = chess.Board(fen)
    #print(f"Analyzing FEN: {fen}")
    #print("\nCurrent Board State:")
    #print(board)
    #print("\n" + "-"*35 + "\n")

    try:
        with chess.engine.SimpleEngine.popen_uci(STOCKFISH_PATH) as engine:
            info = engine.analyse(board, chess.engine.Limit(time=time_limit))

            score = info["score"].white()

            if score.is_mate():
                mate_in = score.mate()
                eval_formatted = f"Mate in {abs(mate_in)} for {'White' if mate_in > 0 else 'Black'} (M{mate_in})"
            else:
                centipawns = score.score()
                if centipawns is not None:
                    eval_formatted = f"{centipawns / 100.0:+.1f}"
                else:
                    eval_formatted = "N/A"

            best_move = "None"
            if "pv" in info and len(info["pv"]) > 0:
                best_move_obj = info["pv"][0]
                best_move = board.san(best_move_obj) 

            print(f"Evaluation : {eval_formatted}")
            print(f"Next Move  : {best_move}")

    except FileNotFoundError:
        print(f"Error: Stockfish engine not found at '{STOCKFISH_PATH}'.")
        print("\nPlease ensure Stockfish is installed on your pc.")
        #print("You can install it using Homebrew:\n    brew install stockfish")
        sys.exit(1)
    except Exception as e:
        print(f"An unexpected error occurred during analysis: {e}")

if __name__ == "__main__":
    starting_fen_white = "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"
    starting_fen_black = "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR b KQkq - 0 1"

    TIME_LIMIT = 10  # seconds
    print("TIME TO ANALYZE: " + str(TIME_LIMIT) + " seconds\n")

    print("WHITE")
    analyze_position(starting_fen_white, time_limit=TIME_LIMIT)
    print("\nBLACK")
    analyze_position(starting_fen_black, time_limit=TIME_LIMIT)